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#  $q$-bic threefolds and their surface of lines

Raymond Cheng  Institute of Algebraic Geometry
Leibniz University Hannover
Welfengarten 1
30167 Hannover
Germany<EMAIL_ADDRESS>
###### Abstract.

For any power $q$ of the positive ground field characteristic, a smooth
$q$-bic threefold—the Fermat threefold of degree $q+1$ for example—has a
smooth surface $S$ of lines which behaves like the Fano surface of a smooth
cubic threefold. I develop projective, moduli-theoretic, and degeneration
techniques to study the geometry of $S$. Using, in addition, the modular
representation theory of the finite unitary group and the geometric theory of
filtrations, I compute cohomology of the structure sheaf of $S$ when $q$ is
prime.

## Introduction

The Fermat threefold of degree $q+1$ is the perhaps the most familiar example
of a _$q$ -bic threefold_, that is, a hypersurface in projective $4$-space of
the form

$X\coloneqq\Set{(x_{0}:x_{1}:x_{2}:x_{3}:x_{4})\in\mathbf{P}^{4}:\sum\nolimits_{i,j=0}^{4}a_{ij}x_{i}^{q}x_{j}=0}$

where $q$ is a power of the ground field characteristic $p>0$. Such
hypersurfaces have long been of recurring interest due to, for example:
idiosyncrasies in their differential projective geometry [Wal56, Hef85, Bea90,
KP91, Nom95]; their abundance of algebraic cycles and relation to
supersingularity [Wei49, Tat65, HM78, SK79, Shi01]; and their unirationality
and geometry of rational curves [Shi74, Con06, BDE+13, She12]. They
classically arise in relation to finite Hermitian geometries [BC66, Seg65,
Hir79, Hir85]; the study of unitary Shimura varieties [Vol10, LTX+22, LZ22];
and Deligne–Lusztig theory for finite unitary groups [Lus76, DL76, Han92,
Li23]. Such hypersurfaces are distinguished in [KKP+22] from the point of view
of singularity theory as those with the lowest possible $F$-pure threshold.
See [Che22, pp. 7–11] for a more extensive survey.

The purpose of this work is to further develop a striking analogy between the
geometry of lines in $q$-bic and cubic hypersurfaces, and also to suggest
flexible framework with which to understand some of the resulting phenomena.
To explain, let $\mathbf{k}$ be, here and throughout, an algebraically closed
field of characteristic $p>0$, and $q$ a fixed power of $p$. Let $X$ be a
smooth $q$-bic threefold over $\mathbf{k}$, and write $S$ for its Fano scheme
of lines. The main results of [Che23b], specialized to dimension $3$, are:

######  Theorem. —

The scheme $S$ of lines of a smooth $q$-bic threefold $X$ is an irreducible,
smooth, projective surface of general type. The Fano correspondence
$S\leftarrow\mathbf{L}\rightarrow X$ induces purely inseparable isogenies

$\mathbf{Alb}_{S}\xrightarrow{\mathbf{L}_{*}}\mathbf{Ab}_{X}^{2}\xrightarrow{\mathbf{L}^{*}}\mathbf{Pic}_{S,\mathrm{red}}^{0}$

amongst supersingular abelian varieties of dimension
$\frac{1}{2}q(q-1)(q^{2}+1)$.

Here, $\mathbf{Ab}_{X}^{2}$ is the _intermediate Jacobian_ of $X$, taken to be
the algebraic representative for algebraically trivial $1$-cycles in $X$, in
the sense of Samuel and Murre, see [Sam60, Bea77, Mur85]; see [Che23b, 6.9]
for a resumé. The statement is analogous to the classical result [CG72,
Theorem 11.19] of Clemens and Griffiths, which states that the Hodge-theoretic
intermediate Jacobian of a complex cubic threefold is isomorphic to the
Albanese variety of its Fano surface of lines. The geometry underlying the
proofs also shares many analogies; compare especially with [Huy23, Chapter 5].
Note that when $q=2$, $X$ is a cubic threefold in characteristic $2$, and this
gives a version of the theorem of Clemens and Griffiths which is, in fact,
new.

The Fano scheme $S$ is the primary object of study in this work, and is
henceforth referred to as the _Fano surface_ of lines of $X$. To begin,
refining the general techniques developed in [Che23b] gives explicit
computations of some basic invariants of $S$. In the following statement,
$\mathcal{S}$ denotes the tautological rank $2$ subbundle, and
$\mathcal{O}_{S}(1)$ is the Plücker line bundle on $S$.

######  Theorem A. —

The tangent bundle $\mathcal{T}_{S}$ of the Fano surface $S$ of a smooth
$q$-bic threefold $X$ is isomorphic to
$\mathcal{S}\otimes\mathcal{O}_{S}(2-q)$. Basic numerical invariants of $S$
are

$\begin{gathered}c_{1}(S)^{2}=(q+1)^{2}(q^{2}+1)(2q-3)^{2},\;\;\;\;c_{2}(S)=(q+1)^{2}(q^{4}-3q^{3}+4q^{2}-4q+3),\;\text{and}\\\
\chi(S,\mathcal{O}_{S})=\frac{1}{12}(q+1)^{2}(5q^{4}-15q^{3}+17q^{2}-16q+12).\end{gathered}$

If $q>2$, then $S$ lifts neither to the second Witt vectors nor to
characteristic $0$.

This is an amalgamation of 1.11, 1.12, and 1.13. The identification of
$\mathcal{T}_{S}$ is an analogue of the tangent bundle theorem [CG72,
Proposition 12.31] of Clemens and Griffiths, see also [AK77, Theorem 1.10].
That $S$ does not lift to the Witt vectors is because it violates
Kodaira–Akizuki–Nakano vanishing, and not to characteristic $0$ because it
violates the Bogomolov–Miyaoka–Yau inequality $c_{1}(S)^{2}\leq 3c_{2}(S)$. As
such, $S$ is a purely positive characteristic phenomenon, and the analogy with
cubics serves as a valuable guide in a setting where classical geometric
intuitions are less potent.

Non-liftability also means that few vanishing theorems are available,
presenting a major difficulty in coherent cohomology computations.
Nonetheless, when $q$ is the prime $p$ itself, cohomology of the structure
sheaf $\mathcal{O}_{S}$ behaves as well as possible:

######  Theorem B. —

Let $X$ be a smooth $q$-bic threefold and $S$ its Fano surface. If $q=p$, then

$\displaystyle\dim_{\mathbf{k}}\mathrm{H}^{1}(S,\mathcal{O}_{S})$
$\displaystyle=\frac{1}{2}p(p-1)(p^{2}+1)=\frac{1}{2}\dim_{\mathbf{Q}_{\ell}}\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(S,\mathbf{Q}_{\ell}),\;\text{and}$
$\displaystyle\dim_{\mathbf{k}}\mathrm{H}^{2}(S,\mathcal{O}_{S})$
$\displaystyle=\frac{1}{12}p(p-1)(5p^{4}-2p^{2}-5p-2).$

In particular, the Picard scheme of $S$ is smooth.

First cohomology is computed in 7.7, at which point second cohomology is
determined by the Euler characteristic computation above. This result,
together with the methods developed, provides a first step towards
understanding the precise relationship amongst the abelian varieties
$\mathbf{Alb}_{S}$, $\mathbf{Ab}_{X}^{2}$, and
$\mathbf{Pic}^{0}_{S,\mathrm{red}}=\mathbf{Pic}^{0}_{S}$ appearing in the
analogue of the Clemens–Griffiths Theorem quoted above.

The method involves a delicate degeneration argument. General considerations
show that the dimension of $\mathrm{H}^{1}(S,\mathcal{O}_{S})$ is at least
half the first Betti number. A corresponding upper bound might be obtained via
upper semicontinuity as follows: Specialize $S$ to a singular surface $S_{0}$
by specializing $X$ to a $q$-bic threefold $X_{0}$ with the mildest possible
singularities; precisely, in terms of the classification of [Che23a, Theorem
A], $X_{0}$ is of type $\mathbf{1}^{\oplus 3}\oplus\mathbf{N}_{2}$, and there
is a choice of coordinates such that

$X_{0}=\Set{(x_{0}:x_{1}:x_{2}:x_{3}:x_{4})\in\mathbf{P}^{4}:x_{0}^{q+1}+x_{1}^{q+1}+x_{2}^{q+1}+x_{3}^{q}x_{4}=0}.$

Although $S_{0}$ is quite singular, its normalization is quite manageable. The
result is as follows:

######  Theorem C. —

The Fano surface $S_{0}$ of a $q$-bic threefold $X_{0}$ of type
$\mathbf{1}^{\oplus 3}\oplus\mathbf{N}_{2}$ is nonnormal, and its
normalization $S_{0}^{\nu}$ is a projective bundle over a smooth $q$-bic curve
$C$. There is a commutative diagram

${S_{0}^{\nu}}$${S_{0}}$${C}$${C}$$\scriptstyle{\nu}$$\scriptstyle{\tilde{\varphi}_{+}}$$\scriptstyle{\varphi_{-}}$$\scriptstyle{\phi_{C}}$

where $\phi_{C}\colon C\to C$ is conjugate to the $q^{2}$-power geometric
Frobenius morphism of $C$. If $q=p$, then

$\displaystyle\dim_{\mathbf{k}}\mathrm{H}^{1}(S_{0},\mathcal{O}_{S_{0}})$
$\displaystyle=\frac{1}{2}p(p-1)(p^{2}+1)+\frac{1}{6}p(p-1)(p-2),\;\;\text{and}$
$\displaystyle\dim_{\mathbf{k}}\mathrm{H}^{2}(S_{0},\mathcal{O}_{S_{0}})$
$\displaystyle=\frac{1}{12}p(p-1)(5p^{4}-2p^{2}-5p-2)+\frac{1}{6}p(p-1)(p-2).$

This collects 4.8, 4.13, and 6.12. See also 6.13 for remarks about extending
the calculation to general $q$. The normalization $S_{0}^{\nu}$ is
constructed, roughly, as the space of lines in $X_{0}$ that project to a
tangent of $C$. Cohomology of $\mathcal{O}_{S_{0}}$ is determined by using the
diagram to relate it to a vector bundle computation $C$; using group schemes
$\mathbf{G}_{m}$ and $\boldsymbol{\alpha}_{p}$ in the automorphism group of
$S_{0}$ to reduce to a manageable set of computations; and using modular
representation theory of the finite unitary group $\mathrm{U}_{3}(p)$,
essentially the automorphism group of $C$, to make explicit identifications.

Alarmingly, upon comparing Theorems B and C, one finds that the cohomology
groups of the special fibre $S_{0}$ are much larger than one would hope. Upper
semicontinuity therefore only shows

$\frac{1}{2}p(p-1)(p^{2}+1)\leq\dim_{\mathbf{k}}\mathrm{H}^{1}(S,\mathcal{O}_{S})\leq\frac{1}{2}p(p-1)(p^{2}+1)+\frac{1}{6}p(p-1)(p-2).$

The upper bound can be refined by carefully analysing a specially chosen
degeneration $S\rightsquigarrow S_{0}$. Specifically, there exists a flat
family $\mathfrak{S}\to\mathbf{A}^{1}$ in which all fibres away from the
central fibre $S_{0}$ are isomorphic to $S$, and which carries a
$\mathbf{G}_{m}$ action compatible with the weight $q^{2}-1$ homothety on the
base $\mathbf{A}^{1}$. The geometric theory of filtrations—the correspondence
between filtrations and $\mathbf{G}_{m}$-equivariant objects over
$\mathbf{A}^{1}$ as pioneered in Simpson’s work [Sim91] on nonabelian Hodge
theory—endows $\mathrm{H}^{1}(S,\mathcal{O}_{S})$ with a filtration whose
graded pieces are various graded pieces of
$\mathrm{H}^{1}(S_{0},\mathcal{O}_{S_{0}})$. A careful analysis identifies
$\frac{1}{6}p(p-1)(p-2)$ cohomology classes on $S_{0}$ that do _not_ lift to
classes on $S$. With this, the upper semicontinuity bound becomes tight,
completing the calculation. This use of the geometric theory of filtrations in
determining cohomology appears to be new, and I expect that the method can be
generalized and refined to be applied in other settings.

Crucial to the degeneration argument is the construction of a fibration
$\varphi\colon S\to C$ on the smooth Fano surface that specializes to
$\varphi_{-}\colon S_{0}\to C$ as in Theorem C. This is done using a general
projective geometry method, which may be adapted to other settings, exploiting
the presence of _cone points_ : points at which the tangent space intersects
the hypersurface $X$ at a cone over a curve $C$; these generalize _Eckardt
points_ of cubic hypersurfaces, and are sometimes also called _star points_. A
combination of projection and intersection with a hyperplane induces a
rational map $S\dashrightarrow C$, which can be resolved in this setting using
global methods made possible by the intrinsic theory of $q$-bic forms
developed in [Che23a]. A summary of the geometric situation is as follows:

######  Theorem D. —

Let $X$ be a $q$-bic threefold either smooth or of type $\mathbf{1}^{\oplus
3}\oplus\mathbf{N}_{2}$, and $S$ its Fano surface. A choice of cone point in
$X$ over a smooth $q$-bic curve $C$ induces a canonical diagram

${\tilde{S}}$${S}$${T}$${C}$$\scriptstyle{\rho}$$\scriptstyle{b}$$\scriptstyle{\varphi}$$\scriptstyle{\pi}$

where
$\varphi_{*}\mathcal{O}_{S}\cong\pi_{*}\mathcal{O}_{T}\cong\mathcal{O}_{C}$,
$b\colon\tilde{S}\to S$ is a blowup along $q^{3}+1$ smooth points, and
$\rho\colon\tilde{S}\to T$ is a quotient by either $\mathbf{F}_{q}$ or
$\boldsymbol{\alpha}_{q}$ depending on whether $X$ is smooth or singular.
Furthermore, $T$ is an explicit rank $1$ degeneracy locus in a product of
projective bundles over $C$.

This is a summary of 2.5, 2.10, 3.4, 3.5, 3.7, 3.9, and 3.10. The techniques
developed here, and more generally in [Che22, Appendix A], may be adapted to
study linear spaces in other projective varieties. For instance, projection
from a cone point maps the scheme of $r$-planes in an $n$-fold to the scheme
of $(r-1)$-planes in an $(n-2)$-fold, providing an inductive method to study
Fano schemes. This method gives a geometric construction of some classically
known structures, such as the fibration constructed in [Rou11, §3.2].

To close the Introduction, I would like to sketch a perspective with which to
contextualize the results at hand. As the prime power $q$ varies, $q$-bic
hypersurfaces are of different degrees, and are even defined over different
characteristics. Yet in many respects, they ought to be viewed as constituting
a single family. One way to make sense of this might be to say that they are
constructed “uniformly with respect to $q$.” This may be formulated in terms
of the shape of the defining equations, but perhaps a more flexible and
geometric way is to realize a $q$-bic hypersurface $X$ as the intersection

$X=\Gamma_{q}\cap Z\subset\mathbf{P}^{n}\times_{\mathbf{k}}\mathbf{P}^{n}$

between the graph $\Gamma_{q}$ of the $q$-power Frobenius morphism, and a
$(1,1)$-divisor $Z$. The Fano surface $S$ also fits into this point of view:
take the ambient variety to be the Grassmannian $\mathbf{G}$, and take $Z$ to
be the zero locus in $\mathbf{G}\times_{\mathbf{k}}\mathbf{G}$ of a general
section of the vector bundle $\mathcal{S}^{\vee}\boxtimes\mathcal{S}^{\vee}$.

This point of view highlights the prime power $q$ and the scheme $Z$ as the
parameters of construction. What should it mean that varying $q$ yields
schemes of the same family? Consider the more familiar matter of fixing $q$
and varying $Z$: Traditionally, those deformations of $X$ or $S$ that are of
the same family are those obtained by _flatly_ varying $Z$, and this is
justified in part by constancy of geometric invariants. When varying $q$, it
is generally too much to ask for invariants to remain constant, but one could
hope that invariants vary in a simple way with respect to $q$. One
particularly pleasant, albeit optimistic, condition would be to ask for Euler
characteristics or, stronger, for dimensions of cohomology groups to vary as a
polynomial in $q$. This is true of $q$-bic hypersurfaces, and Theorems B and C
may be understood as saying the stronger condition of constancy of
cohomological invariants holds also for the Fano surfaces of $q$-bic
threefolds, at least when $q$ varies only over primes $p$. I believe it is an
interesting problem to find more examples of this type of phenomenon.

Finally, observe that if $\mathbf{P}^{n}$ and $\mathbf{G}$ were to be replaced
by more general ambient varieties $\mathrm{P}$, and $Z$ allowed to be
subschemes not necessarily cut out by sections of vector bundles, even Euler
characteristics cannot vary as simply as a polynomial in $q$: indeed, by
taking $Z$ to be the diagonal, this would count points of $\mathrm{P}$ over
finite fields. However, the Weil conjectures show that, even then, there is
structure amongst the numbers and, taken together, the give insight into the
geometry of $\mathrm{P}$. My hope is that, by circumscribing the class of
ambient schemes $\mathrm{P}$ and subschemes $Z$, this line of study may lead
to new and interesting structures in positive characteristic geometry.

Outline. — §1 begins with a summary of the theory of $q$-bic forms and
hypersurfaces, as developed in [Che23a, Che23b]; the latter half refines these
methods in order to establish Theorem A. §§2–3 constructs and studies a
rational map $S\dashrightarrow C$ from the Fano surface to a $q$-bic curve,
establishing Theorem D. These methods are applied to study $q$-bic threefolds
of type $\mathbf{1}^{\oplus 3}\oplus\mathbf{N}_{2}$ in §4. §§5–6 are where
most of the cohomology computation for singular $q$-bics takes place, and
completes the proof of Theorem C. Finally, §7 puts everything together to
prove Theorem B.

Acknowledgements. — This article is based on parts of Chapter 4 of my thesis
[Che22]. Sincerest gratitude goes to Aise Johan de Jong for many conversations
around half-ideas and failed attempts that went into this work. Thanks to
Matthias Schütt for comments on early drafts. During the completion of this
work, I was supported by a Humboldt Research Fellowship.

##  1\. $q$-bic hypersurfaces

The first half of this section summarizes some of the theory and results
developed in [Che23a, Che23b] regarding $q$-bic forms and $q$-bic
hypersurfaces that will be freely used in this article. The second half of
this section, starting in 1.10, develops a few basic results pertaining to the
Fano surface of a $q$-bic threefold. Throughout this article, denote by
$\mathbf{k}$ an algebraically closed field of characteristic $p>0$,
$q\coloneqq p^{e}$ for some $e\in\mathbf{Z}_{>0}$, and $V$ a
$\mathbf{k}$-vector space of dimension $n+1$.

###  1.1. $q$-bic forms and hypersurfaces

— A _$q$ -bic form_ on $V$ is a nonzero linear map

$\beta\colon V^{[1]}\otimes_{\mathbf{k}}V\to\mathbf{k},$

where here and elsewhere,
$V^{[1]}\coloneqq\mathbf{k}\otimes_{\operatorname{Fr},\mathbf{k}}V$ is the
$q$-power Frobenius twist of $V$. Let $\mathbf{P}V$ be the projective space of
lines associated with $V$. The _$q$ -bic equation_ associated with $\beta$ is
the map

$f_{\beta}\coloneqq\beta(\mathrm{eu}^{[1]},\mathrm{eu})\colon\mathcal{O}_{\mathbf{P}V}(-q-1)\to
V^{[1]}\otimes_{\mathbf{k}}V\otimes_{\mathbf{k}}\mathcal{O}_{\mathbf{P}V}\to\mathcal{O}_{\mathbf{P}V}$

obtained by pairing the canonical section
$\mathrm{eu}\colon\mathcal{O}_{\mathbf{P}V}(-1)\to
V\otimes_{\mathbf{k}}\mathcal{O}_{\mathbf{P}V}$ with its $q$-power via
$\beta$. The _$q$ -bic hypersurface_ associated with the $q$-bic form $\beta$
is the hypersurface defined by $f_{\beta}$, and may be identified as the space
of lines in $V$ isotropic for $\beta$:

$X\coloneqq
X_{\beta}\coloneqq\mathrm{V}(f_{\beta})=\set{[v]\in\mathbf{P}V:\beta(v^{[1]},v)=0}.$

A choice of basis $V=\langle e_{0},\ldots,e_{n}\rangle$ makes this more
explicit: Writing $\mathbf{x}^{\vee}\coloneqq(x_{0}:\cdots:x_{n})$ for the
corresponding projective coordinates on $\mathbf{P}V\cong\mathbf{P}^{n}$, and
$a_{ij}\coloneqq\beta(e_{i}^{[1]},e_{j})$ for the $(i,j)$-th entry of the Gram
matrix of $\beta$ with respect to the given basis, as in [Che23a, 1.2], the
$q$-bic equation $f_{\beta}$ is the homogeneous polynomial of degree $q+1$
given by

$f_{\beta}(x_{0},\ldots,x_{n})\coloneqq\mathbf{x}^{[1],\vee}\cdot\operatorname{Gram}(\beta;e_{0},\ldots,e_{n})\cdot\mathbf{x}=\sum\nolimits_{i,j=0}^{n}a_{ij}x_{i}^{q}x_{j}.$

From any of the descriptions, it is easy to check that a hyperplane section of
a $q$-bic hypersurface is a $q$-bic hypersurface: see [Che23b, 1.9].

To illustrate the utility of distinguishing the shape of the $q$-bic equation
$f\coloneqq f_{\beta}$, and to record the computation for later use, the
following shows that the action of the $q$-power absolute Frobenius morphism
$\operatorname{Fr}\colon X\to X$ is zero on higher cohomology:

######  1.2 Lemma. —

If $\dim X\geq 1$, then the map
$\operatorname{Fr}\colon\mathrm{H}^{n-1}(X,\mathcal{O}_{X})\to\mathrm{H}^{n-1}(X,\mathcal{O}_{X})$
is zero.

###### Proof.

Consider the commutative diagram of abelian sheaves on $\mathbf{P}V$ given by

${0}$${\mathcal{O}_{\mathbf{P}V}(-q-1)}$${\mathcal{O}_{\mathbf{P}V}}$${\mathcal{O}_{X}}$${0}$${0}$${\operatorname{Fr}_{*}(\mathcal{O}_{\mathbf{P}V}(-q-1))}$${\operatorname{Fr}_{*}\mathcal{O}_{\mathbf{P}V}}$${\operatorname{Fr}_{*}\mathcal{O}_{X}}$${0}$$\scriptstyle{f}$$\scriptstyle{f^{q-1}\operatorname{Fr}}$$\scriptstyle{\operatorname{Fr}}$$\scriptstyle{\operatorname{Fr}}$$\scriptstyle{f^{q}}$

where $\operatorname{Fr}$ acts on local sections by $s\mapsto s^{q}$. Taking
cohomology yields a commutative diagram

${\mathrm{H}^{n-1}(X,\mathcal{O}_{X})}$${\mathrm{H}^{n}(\mathbf{P}V,\mathcal{O}_{\mathbf{P}V}(-q-1))}$${\mathrm{H}^{n-1}(X,\mathcal{O}_{X})}$${\mathrm{H}^{n}(\mathbf{P}V,\mathcal{O}_{\mathbf{P}V}(-q-1))\makebox[0.0pt][l]{\,.}}$$\scriptstyle{\cong}$$\scriptstyle{\operatorname{Fr}}$$\scriptstyle{f^{q-1}\operatorname{Fr}}$$\scriptstyle{\cong}$

Compute the map $f^{q-1}\operatorname{Fr}$ on the right as follows: Upon
choosing coordinates $(x_{0}:\cdots:x_{n})$ for $\mathbf{P}V=\mathbf{P}^{n}$,
a basis for $\mathrm{H}^{n}(\mathbf{P}V,\mathcal{O}_{\mathbf{P}V}(-q-1))$ is
given by elements

$\xi\coloneqq(x_{0}^{i_{0}}\cdots
x_{n}^{i_{n}})^{-1}\;\;\text{where}\;i_{0}+\cdots+i_{n}=q+1\;\text{and
each}\;i_{j}\geq 1.$

Monomials appearing in $f^{q-1}$ are of the form $a^{q}b$, where $a$ and $b$
are themselves monomials of degree $q-1$. Write $a=x_{0}^{a_{0}}\cdots
x_{n}^{a_{n}}$ with $a_{0}+\cdots+a_{n}=q-1$. Then
$f^{q-1}\operatorname{Fr}(\xi)$ is a sum of terms

$a^{q}b\cdot(x_{0}^{i_{0}}\cdots
x_{n}^{i_{n}})^{-q}=b\cdot(x_{0}^{i_{0}-a_{0}}\cdots
x_{n}^{i_{n}-a_{n}})^{-q}.$

Since $(i_{0}-a_{0})+\cdots+(i_{n}-a_{n})=2\leq n$, there is some index $j$
such that $i_{j}-a_{j}\leq 0$, and so this represents $0$. Thus each potential
contribution to $f^{q-1}\operatorname{Fr}(\xi)$ vanishes, and so
$f^{q-1}\operatorname{Fr}(\xi)=0$. ∎

###  1.3. Classification

— Isomorphic $q$-bic forms yield projectively equivalent $q$-bic
hypersurfaces. Over an algebraically closed field $\mathbf{k}$, [Che23a,
Theorem A] shows that there are finitely many isomorphism classes of $q$-bic
forms on $V$, and that they may be classified via their Gram matrices: there
exists a basis $V=\langle e_{0},\ldots,e_{n}\rangle$ and nonnegative integers
$a,b_{m}\in\mathbf{Z}_{\geq 0}$ such that

$\operatorname{Gram}(\beta;e_{0},\ldots,e_{n})=\mathbf{1}^{\oplus
a}\oplus\Big{(}\bigoplus\nolimits_{m\geq 1}\mathbf{N}_{m}^{\oplus
b_{m}}\Big{)}$

where $\mathbf{1}$ is the $1$-by-$1$ matrix with unique entry $1$,
$\mathbf{N}_{m}$ is the $m$-by-$m$ Jordan block with $0$’s on the diagonal,
and $\oplus$ denotes block diagonal sums of matrices. The tuple
$(a;b_{m})_{m\geq 1}$ is called the _type_ of $\beta$, and is the fundamental
invariant of; the sum $\sum_{m\geq 1}b_{m}$ is its _corank_. When $q$-bic
forms are varied in a family, the corank can jump and the type can change; the
Hasse diagram of specialization relations is not linear in general, and some
partial relations are given in [Che23a, Theorem C].

Of note is the $\mathbf{N}_{1}^{\oplus b_{1}}$ piece, which is a
$b_{1}$-by-$b_{1}$ zero matrix, and will sometimes denoted by
$\mathbf{0}^{\oplus b_{1}}$. Its underlying subspace is the _radical_ of
$\beta$:

$\operatorname{rad}(\beta)=\set{v\in
V:\beta(v^{[1]},u)=\beta(u^{[1]},v)=0\;\text{for all}\;u\in V}.$

It is straightforward to check that $X$ is a cone if and only if $b_{1}\neq
0$, and that the vertex is the projectivization of the radical; see also
[Che22, 2.4] for an invariant treatment.

###  1.4. $q$-bic points and curves

— In low-dimensions, the classification and specialization relations are quite
simple and, especially when combined with the fact that hyperplane sections of
$q$-bics are $q$-bics, are geometrically very useful: see [Che22, Chapter 3]
for more. When $V$ has dimension $2$, the possible types of $q$-bic forms and
their specialization relations are

$\mathbf{1}^{\oplus
2}\rightsquigarrow\mathbf{N}_{2}\rightsquigarrow\mathbf{0}\oplus\mathbf{1},$

corresponding to the subschemes in $\mathbf{P}^{1}$ defined by
$x_{0}^{q+1}+x_{1}^{q+1}$, $x_{0}^{q}x_{1}$, and $x_{1}^{q+1}$. Thus $q$-bic
points are either $q+1$ reduced points, a reduced point with a $q$-fold point,
or one point of multiplicity $q+1$.

When $V$ is $3$-dimensional, the types and specializations excluding cones
over $q$-bic points are

$\mathbf{1}^{\oplus
3}\rightsquigarrow\mathbf{1}\oplus\mathbf{N}_{2}\rightsquigarrow\mathbf{N}_{3}.$

The corresponding subschemes of $\mathbf{P}^{2}$ may be described as: a smooth
curve of degree $q+1$; an irreducible, geometrically rational curve with a
single unibranch singularity; and a reducible curve with a linear component
meeting an irreducible, geometrically rational component of degree $q$ at its
unique unibranch singularity.

###  1.5. Smoothness

— A $q$-bic hypersurface $X$ is smooth if and only if its underlying $q$-bic
form $\beta$ is _nonsingular_ in the sense that one—equivalently, both—of the
adjoint maps $\beta\colon V\to V^{[1],\vee}$ or $\beta^{\vee}\colon V^{[1]}\to
V^{\vee}$ is an isomorphism; with a choice of basis, this is equivalent to
invertibility of a Gram matrix. The singular locus of $X$ is supported on the
projectivization of the subspace

$V^{\perp,[-1]}\coloneqq\set{v\in V:\beta(v^{[1]},u)=0\;\text{for every}\;u\in
V},$

the Frobenius descent of the left kernel
$V^{\perp}\coloneqq\ker(\beta^{\vee}\colon V^{[1]}\to V^{\vee})$: see [Che23b,
2.6]. The embedded tangent space $\mathbf{T}_{X,x}\subset\mathbf{P}V$ to $X$
at a point $x=\mathbf{P}L$ is the projectivization of

$L^{[1],\perp}\coloneqq\set{v\in V:\beta(u^{[1]},v)=0\;\text{for all}\;u\in
L}\coloneqq\ker(\beta\colon V\to V^{[1],\vee}\twoheadrightarrow
L^{[1],\vee}),$

the right orthogonal of $L$: see [Che23b, 2.2].

###  1.6. Hermitian structures

— A vector $v\in V$ is said to be _Hermitian_ if

$\beta(u^{[1]},v)=\beta(v^{[1]},u)^{q}\;\;\text{for all}\;u\in V.$

A subspace $U\subset V$ is _Hermitian_ if it is spanned by Hermitian vectors.
The Hermitian equation implies that the left and right orthogonals of $U$,

$\displaystyle U^{\perp,[-1]}$ $\displaystyle\coloneqq\set{v\in
V:\beta(v^{[1]},u)=0\;\text{for all}\;u\in U}\;\text{and}$ $\displaystyle
U^{[1],\perp}$ $\displaystyle\coloneqq\set{v\in
V:\beta(u^{[1]},v)=0\;\text{for all}\;u\in U},$

coincide. When $\beta$ is nonsingular, there are only finitely many Hermitian
vectors, whence finitely many Hermitian subspaces, and they span $V$: see
[Che23a, 2.1–2.6]

Continuing with $\beta$ nonsingular, there is a canonical $\mathbf{k}$-vector
space isomorphism

$\sigma_{\beta}\coloneqq\beta^{-1}\circ\beta^{[1],\vee}\colon V^{[2]}\to
V^{[1],\vee}\to V$

between the $q^{2}$-Frobenius twist of $V$ and $V$ itself; this is an
$\mathbf{F}_{q^{2}}$ descent datum on $V$. Pre-composing with the universal
$q^{2}$-linear map $V\to V^{[2]}$ yields a $q^{2}$-linear endomorphism
$\phi\colon V\to V$. It preserves isotropicity so it induces an endomorphism
$\phi_{X}\colon X\to X$. In a basis $V=\langle e_{0},\ldots,e_{n}\rangle$ of
Hermitian vectors, the Gram matrix of $\beta$ is Hermitian in the sense that

$\operatorname{Gram}(\beta;e_{0},\ldots,e_{n})^{\vee}=\operatorname{Gram}(\beta;e_{0},\ldots,e_{n})^{[1]},$

from which it follows that $\sigma_{\beta}$ is represented by the identity
matrix, and that $\phi_{X}$ is the $q^{2}$-power geometric Frobenius morphism
in the corresponding coordinates: see [Che23b, 4.3].

By [Che23b, 1.3], the fixed set of $\phi_{X}$ is the subset
$X_{\mathrm{Herm}}$ of _Hermitian points_ of $X$, consisting of those points
$x=\mathbf{P}L$ where $L$ is spanned by a Hermitian vector. Comparing with the
construction of $\phi_{X}$, this implies that the set of Hermitian points of
$X$ is the zero locus of the map

$\tilde{\theta}\colon\mathcal{O}_{X}(-q^{2})\stackrel{{\scriptstyle\mathrm{eu}^{[2]}}}{{\longrightarrow}}V^{[2]}\otimes_{\mathbf{k}}\mathcal{O}_{X}\stackrel{{\scriptstyle\sigma_{\beta}}}{{\longrightarrow}}V\otimes_{\mathbf{k}}\mathcal{O}_{X}\longrightarrow\mathcal{T}_{\mathbf{P}V}(-1)\rvert_{X}$

where the final arrow arises from the dual Euler sequence. The following shows
that this map factors through the tangent bundle of $X$. A geometric
description of $\phi_{X}(x)$ may be extracted from the proof below, as can be
found in [Che22, 2.9.9], and this is most clear when $X$ is a curve: the
tangent line $\mathbf{T}_{X,x}$ intersects $X$ at $x$ with multiplicity $q$,
and the residual point of intersection is $\phi_{X}(x)$.

######  1.7 Lemma. —

Hermitian points of $X$ are cut out by a canonical map
$\theta\colon\mathcal{O}_{X}(-q^{2})\to\mathcal{T}_{X}(-1)$.

###### Proof.

The tangent bundle of $X$ is the kernel of the normal map
$\delta\colon\mathcal{T}_{\mathbf{P}V}(-1)\rvert_{X}\to\mathcal{N}_{X/\mathbf{P}V}(-1)$;
this is induced via the dual Euler sequence by the map $\tilde{\delta}\colon
V\otimes_{\mathbf{k}}\mathcal{O}_{X}\to\mathcal{N}_{X/\mathbf{P}V}(-1)$ which
takes directional derivatives of a fixed equation of $X$. A simple computation
shows that there is a commutative diagram

${V\otimes\mathcal{O}_{X}}$${\mathcal{N}_{X/\mathbf{P}V}(-1)}$${V^{[1],\vee}\otimes\mathcal{O}_{X}}$${\mathcal{O}_{X}(q)}$$\scriptstyle{\tilde{\delta}}$$\scriptstyle{\beta}$$\scriptstyle{\mathrm{eu}^{[1],\vee}}$$\scriptstyle{\cong}$

and so $\delta\circ\tilde{\theta}$ factors through the map

$\tilde{\delta}\circ\sigma_{\beta}\circ\mathrm{eu}^{[2]}=\mathrm{eu}^{[1],\vee}\circ\beta^{[1],\vee}\circ\mathrm{eu}^{[2]}=\beta(\mathrm{eu}^{[1]},\mathrm{eu})^{q}=0,$

see [Che23a, 1.7] for the pentultimate relation. This implies that
$\tilde{\theta}$ factors through $\mathcal{T}_{X}(-1)$. ∎

###  1.8. Scheme of lines

— The Fano scheme $\mathbf{F}\coloneqq\mathbf{F}_{1}(X)$ of lines of a $q$-bic
hypersurface $X$ may be viewed as the space of $2$-dimensional subspaces in
$V$ which are totally isotropic for the $q$-bic form $\beta$:

$\mathbf{F}=\set{[U]\in\mathbf{G}:\beta(u^{[1]},u^{\prime})=0\;\text{for
all}\;u,u^{\prime}\in U}.$

This description exhibits $\mathbf{F}$ as the zero locus in the Grassmannaian
$\mathbf{G}\coloneqq\mathbf{G}(2,V)$ of the map

$\beta_{\mathcal{S}}\colon\mathcal{S}^{[1]}\otimes_{\mathcal{O}_{\mathbf{G}}}\mathcal{S}\to\mathcal{O}_{\mathbf{G}}$

induced by restricting $\beta$ to the universal rank $2$ subbundle
$\mathcal{S}\subset V\otimes_{\mathbf{k}}\mathcal{O}_{\mathbf{G}}$. A direct
construction shows that $\mathbf{F}$ is nonempty whenever $n\geq 3$, so this
implies that $\dim\mathbf{F}\geq 2n-6$: see [Che23b, 1.10–1.12]. Furthermore,
$\mathbf{F}$ is connected whenever $n\geq 4$: see [Che23b, 2.9].

When $\mathbf{F}$ is generically smooth and of expected dimension $2n-6$, it
is a local complete intersection scheme whose structure sheaf admits a Koszul
resolution by
$\wedge^{*}\mathcal{S}^{[1]}\otimes_{\mathcal{O}_{\mathbf{G}}}\mathcal{S}$,
and its dualizing sheaf may be computed to be

$\omega_{\mathbf{F}}\cong\mathcal{O}_{\mathbf{F}}(2q+1-n)\otimes_{\mathbf{k}}\det(V)^{\vee,\otimes
2}$

where the twist by $\det(V)$ is useful for tracking weights of algebraic group
actions: see [Che23b, 2.4]. Here, $\mathcal{O}_{\mathbf{F}}(1)$ is the Plücker
line bundle; its degree is computed in [Che23b, 1.15] as

$\deg\mathcal{O}_{\mathbf{F}}(1)=\frac{(2n-6)!}{(n-1)!(n-3)!}(q+1)^{2}\big{(}(n-1)q^{2}+(2n-8)q+(n-1)\big{)}.$

The Fano scheme $\mathbf{F}$ is singular along the subscheme of lines through
the singular locus of $X$:

$\operatorname{Sing}\mathbf{F}=\Set{[\ell]\in\mathbf{F}:\ell\cap\operatorname{Sing}X\neq\varnothing}.$

In particular, $\mathbf{F}$ is smooth if and only if $X$ itself is smooth, or
equivalently, when $\beta$ is nonsingular: see [Che23b, 2.7]. This description
moreover implies that $\mathbf{F}$ is of expected dimension if and only if the
locus of lines through the singular locus of $X$ has dimension at most $2n-6$.

###  1.9. $q$-bic surfaces

— To complete the tour in low dimensions, consider a $q$-bic hypersurface $X$
of dimension $2$, or briefly, a _$q$ -bic surface_. Classification and
specialization of types looks like

${\mathbf{1}^{4}}$${\mathbf{N}_{2}\mathbf{1}^{2}}$${\mathbf{N}_{4}}$${\mathbf{0}\mathbf{N}_{2}\mathbf{1}}$${\mathbf{0}\mathbf{N}_{3}}$${\mathbf{0}^{2}\mathbf{1}^{2}}$${\mathbf{0}^{2}\mathbf{N}_{2}}$${\mathbf{0}^{3}\mathbf{1}}$${\mathbf{N}_{3}\mathbf{1}}$${\mathbf{0}\mathbf{1}^{3}}$$\rightsquigarrow$$\rightsquigarrow$$\rightsquigarrow$${\mathbf{N}_{2}^{2}}$${\rightsquigarrow}$${\rightsquigarrow}$$\rightsquigarrow$$\rightsquigarrow$$\rightsquigarrow$$\rightsquigarrow$$\rightsquigarrow$$\rightsquigarrow$

where, notably, the Hasse diagram of specialization relations is not linear.
See [Che22, 3.8] for more.

The general statements of 1.8 shows that $X$ always contains lines, and that
its Fano scheme $\mathbf{F}$ has degree $(q+1)(q^{3}+1)$ when it is of
expected dimension $0$. This means, in particular, that a smooth $q$-bic
surface has exact $(q+1)(q^{3}+1)$ distinct lines. The configuration of these
lines is fascinating and is well-studied: see, for example, [Hir85, SSvL10,
BPRS21]. Other aspects of the geometry of smooth $q$-bic surfaces may be
found, for example, in [Shi74, Oji19].

###  1.10. $q$-bic threefolds

— In the remainder of the article, $X$ denotes a $3$-dimensional $q$-bic
hypersurface, or a _$q$ -bic threefold_ for short. The classification and
specialization relations of $q$-bic threefolds, at least up to the first few
cones, looks as follows:

${\mathbf{1}^{5}}$${\mathbf{1}^{3}\mathbf{N}_{2}}$${\mathbf{1}\mathbf{N}_{4}}$${\mathbf{N}_{5}}$${\mathbf{1}^{2}\mathbf{N}_{3}}$${\mathbf{0}\mathbf{1}^{4}}$${\mathbf{1}\mathbf{N}_{2}^{2}}$${\mathbf{N}_{2}\mathbf{N}_{3}}$${\mathbf{0}\mathbf{1}^{2}\mathbf{N}_{2}\makebox[0.0pt][l]{\,.}}$$\rightsquigarrow$$\rightsquigarrow$$\rightsquigarrow$$\rightsquigarrow$$\rightsquigarrow$$\rightsquigarrow$$\rightsquigarrow$$\rightsquigarrow$$\rightsquigarrow$$\rightsquigarrow$

Notice that all non-cones have corank at most $2$, which by the discussion of
1.5, implies that the singular locus of a $q$-bic threefold which is not a
cone has dimension at most $1$.

The Fano scheme $S\coloneqq\mathbf{F}$ of lines of a $q$-bic threefold has
expected dimension $2$ by 1.8. It is easy to check that when $X$ has corank at
most $1$ and is not a cone, the singular locus of $S$ has dimension at most
$1$. Thus in all such cases, and also in the case $X$ is smooth, $S$ is
generically smooth of expected dimension $2$: see also [Che23b, 2.8]. In any
of these cases, but especially when $X$ is either smooth or of type
$\mathbf{1}^{\oplus 3}\oplus\mathbf{N}_{2}$, $S$ will be referred to as the
_Fano surface_ of $X$.

Begin by considering the Fano surface $S$ of a smooth $q$-bic threefold $X$,
describing the tangent bundle $\mathcal{T}_{S}$, and computing its Chern
numbers:

######  1.11 Lemma. —

If $X$ is smooth, then
$\mathcal{T}_{S}\cong\mathcal{S}\otimes_{\mathcal{O}_{S}}\mathcal{O}_{S}(2-q)$
and the Chern numbers of $S$ are

$c_{1}(S)^{2}=(q+1)^{2}(q^{2}+1)(2q-3)^{2}\;\;\text{and}\;\;c_{2}(S)=(q+1)^{2}(q^{4}-3q^{3}+4q^{2}-4q+3).$

###### Proof.

Identify the tangent bundle of $S$ as in [Che23b, 2.2] as
$\mathcal{T}_{S}\cong\operatorname{\mathcal{H}\\!\mathit{om}}_{\mathcal{O}_{S}}(\mathcal{S},\mathcal{S}^{[1],\perp}/\mathcal{S})$,
where the sheaf in the target of the
$\operatorname{\mathcal{H}\\!\mathit{om}}$ is characterized by the short exact
sequence

$0\to\mathcal{S}^{[1],\perp}/\mathcal{S}\to\mathcal{Q}\xrightarrow{\beta}\mathcal{S}^{[1],\vee}\to
0.$

Taking determinants shows
$\mathcal{S}^{[1],\perp}/\mathcal{S}\cong\mathcal{O}_{S}(1-q)$, and combined
with the wedge product isomorphism
$\mathcal{S}^{\vee}\cong\mathcal{S}\otimes_{\mathcal{O}_{S}}\mathcal{O}_{S}(1)$
gives the identification of $\mathcal{T}_{S}$. To compute the first Chern
number, note that the Plücker line bundle $\mathcal{O}_{S}(1)$ has degree
$(q+1)^{2}(q^{2}+1)$ by taking $n=4$ in the formula given in 1.8. Combined
with the identification of $\mathcal{T}_{S}$, this gives

$c_{1}(S)^{2}\coloneqq\int_{S}c_{1}(\mathcal{T}_{S})^{2}=\int_{S}c_{1}(\mathcal{O}_{S}(3-2q))^{2}=(2q-3)^{2}\deg\mathcal{O}_{S}(1)=(q+1)^{2}(q^{2}+1)(2q-3)^{2}.$

For the second Chern number, observe that

$\displaystyle c_{2}(\mathcal{T}_{S})$
$\displaystyle=c_{2}(\mathcal{S}^{\vee})+c_{1}(\mathcal{S}^{\vee})c_{1}(\mathcal{O}_{S}(1-q))+c_{1}(\mathcal{O}_{S}(1-q))^{2}$
$\displaystyle=c_{2}(\mathcal{S}^{\vee})+(q-2)(q-1)c_{1}(\mathcal{O}_{S}(1))^{2}.$

Since a general section of $\mathcal{S}^{\vee}$ cuts out the scheme of lines
contained in a general hyperplane section of $X$, which is but a smooth
$q$-bic surface by 1.1 and 1.3, the degree of $c_{2}(\mathcal{S}^{\vee})$ is
$(q+1)(q^{3}+1)$ by 1.9. Combining with the degree computation for
$\mathcal{O}_{S}(1)$ gives $c_{2}(S)$. ∎

This computation implies that the Fano surface $S$ cannot lift to either the
second Witt vectors of $\mathbf{k}$ or any characteristic $0$ base as soon as
$q>2$. By contrast, observe that $S$ _does_ lift when $q=2$, as is seen by
taking any lift of $X$ and by applying the Fano scheme construction in
families.

######  1.12 Proposition. —

If $X$ is smooth and $q>2$, then $S$ lifts neither to
$\mathrm{W}_{2}(\mathbf{k})$ nor to characteristic $0$.

###### Proof.

Dualizing the tangent bundle computation of 1.11 shows
$\Omega^{1}_{S}\cong\mathcal{S}^{\vee}\otimes_{\mathcal{O}_{S}}\mathcal{O}_{S}(q-2)$.
Since $\mathcal{S}^{\vee}$ has sections, this implies that
Kodaira–Akizuki–Nakano vanishing fails on $S$, so [DI87, Corollaire 2.8] shows
$S$ cannot lift to $\mathrm{W}_{2}(\mathbf{k})$. The Chern number computation
of 1.11 gives

$c_{1}(S)^{2}-3c_{2}(S)=q^{2}(q+1)^{2}(q^{2}-3q+1).$

The quadratic formula shows that this is positive whenever $q>2$, and so $S$
the Bogomolov–Miyaoka–Yau inequality of [Miy77, Theorem 4] is not satisfied on
$S$. Since Chern numbers are constant in flat families, this implies that $S$
cannot lift to characteristic $0$. ∎

The Chern number computation also gives a simple way to compute the Euler
characteristic of the structure sheaf $\mathcal{O}_{S}$ whenever $S$ is
smooth; this holds more generally, whenever $S$ has expected dimension, by
comparing with the Koszul resolution of $\mathcal{O}_{S}$ on $\mathbf{G}$:

######  1.13 Proposition. —

If $S$ is of its expected dimension $2$, then

$\chi(S,\mathcal{O}_{S})=\frac{1}{12}(q+1)^{2}(5q^{4}-15q^{3}+17q^{2}-16q+12).$

###### Proof.

When $X$ is a smooth, so is $S$, and so Noether’s formula, as in [Ful98,
Example 15.2.2], applies to give the first equality in

$\chi(S,\mathcal{O}_{S})=\frac{1}{12}(c_{1}(S)^{2}+c_{2}(S))=\sum\nolimits_{i=0}^{4}(-1)^{i}\chi(\mathbf{G},\wedge^{i}\mathcal{S}^{[1]}\otimes_{\mathcal{O}_{\mathbf{G}}}\mathcal{S}).$

Substituting the Chern number computations from 1.11 gives the formula in the
statement. The second equality arises from taking the Koszul resolution of
$\mathcal{O}_{S}$ on $\mathbf{G}$. Since the Koszul resolution persists
whenever $S$ is of dimension $2$, this gives the general case. ∎

##  2\. Cone situation

Study the Fano scheme $S$ by fibering it over a $q$-bic curve $C$. This is
easy to arrange rationally, as is promptly explained in 2.5, and the main
content of this section is to construct a canonical resolution of the rational
map $S\dashrightarrow C$ using a mixture of projective geometry techniques and
the intrinsic theory of $q$-bic forms from [Che23a]: see 2.11 for a summary.
The setting in this section is quite general; the most important situations
for the main computations of the article—where additional smoothness
hypotheses are satisfied—are classified in 2.3(ii), and are further studied in
§3.

###  2.1. Cone situation

— A triple $(X,\infty,\mathbf{P}\breve{W})$ consisting of a $q$-bic threefold
$X$, a point $\infty=\mathbf{P}L$ of $X$, and a hyperplane
$\mathbf{P}\breve{W}$ such that $X\cap\mathbf{P}\breve{W}$ is a cone with
vertex $\infty$ over a reduced $q$-bic curve
$C\subset\mathbf{P}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{W}$
is called a _cone situation_ ; here and later, $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{W}\coloneqq\breve{W}/L$ and $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{V}\coloneqq V/L$. This is sometimes considered with some of the following
additional geometric assumptions:

  1. (i)

there does not exist a $2$-plane contained in $X$ which passes through
$\infty$; or

  2. (ii)

$\infty$ is a smooth point of $X$; or

  3. (iii)

$C$ is a smooth $q$-bic curve.

Conditions (ii) and (iii) together imply (i). That $\infty$ is a vertex for
$X\cap\mathbf{P}\breve{W}$ means that $L$ lies in the radical of
$\beta\rvert_{W}$, so that $W$ is contained in both orthogonals of $L$, see
1.3. In particular, this means that $\mathbf{P}\breve{W}$ is contained in the
embedded tangent space $\mathbf{T}_{X,\infty}=\mathbf{P}L^{[1],\perp}$ of $X$
at $\infty$, see 1.5; and if (ii) is satisfied, then equality
$\mathbf{P}\breve{W}=\mathbf{T}_{X,\infty}$ necessarily holds.

###  2.2. Examples

— The following are some examples of cone situations:

  1. (i)

Let $X$ be a smooth $q$-bic threefold and let $\infty\in X$ be a Hermitian
point. Then $(X,\infty,\mathbf{T}_{X,\infty})$ is a cone situation, and it
follows from 1.3 and 1.6 that all cone situations for smooth $X$ are obtained
this way. All the conditions (i), (ii), and (iii) are satisfied.

The next three examples pertain to $q$-bic threefolds of type
$\mathbf{1}^{\oplus 3}\oplus\mathbf{N}_{2}$. For concreteness, let

$X=\mathrm{V}(x_{0}^{q}x_{1}+x_{0}x_{1}^{q}+x_{2}^{q+1}+x_{3}^{q}x_{4})\subset\mathbf{P}^{4},$

and set $C\coloneqq\mathrm{V}(x_{0}^{q}x_{1}+x_{0}x_{1}^{q}+x_{2}^{q+1})$ in
the $\mathbf{P}^{2}$ in which $x_{3}$ and $x_{4}$ are projected out. Write
$x_{-}\coloneqq(0:0:0:1:0)$, $x_{+}\coloneqq(0:0:0:0:1)$, and
$\infty\coloneqq(1:0:0:0:0)$.

  1. (ii)

$(X,x_{-},\mathbf{T}_{X,x_{-}})$ is a cone situation over $C$ satisfying each
of the conditions (i), (ii), and (iii).

  2. (iii)

$(X,x_{+},\mathrm{V}(x_{3}))$ is a cone situation over the $C$ satisfying (i)
and (iii), but not (ii).

  3. (iv)

$(X,\infty,\mathbf{T}_{X,\infty})$ is a cone situation over
$\mathrm{V}(x_{2}^{q+1}+x_{3}^{q}x_{4})$ satisfying (i) and (ii), but not
(iii).

The remaining examples illustrate the range of possibilities for the cone
situation.

  1. (v)

Let $X\coloneqq\mathrm{V}(x_{0}^{q}x_{1}+x_{1}^{q}x_{2}+x_{3}^{q}x_{4})$ and
$\infty\coloneqq(0:0:0:1:0)$. Then $(X,\infty,\mathbf{T}_{X,\infty})$ is a
cone situation over $\mathrm{V}(x_{0}^{q}x_{1}+x_{1}^{q}x_{2})$ satisfying
(ii), but not (i) nor (iii).

  2. (vi)

Let $X$ be of type $\mathbf{0}\oplus\mathbf{1}^{\oplus 4}$, $\infty$ the
vertex of $X$, and $\mathbf{P}\breve{W}$ a general hyperplane through
$\infty$. Then $(X,\infty,\mathbf{P}\breve{W})$ is a cone situation satisfying
(iii), but not (i) nor (ii).

  3. (vii)

Let $X$ be of type $\mathbf{0}^{\oplus
2}\oplus\mathbf{1}\oplus\mathbf{N}_{2}$, $\infty$ any point of the vertex of
$X$, and $\mathbf{P}\breve{W}$ be any hyperplane intersecting the vertex of
$X$ exactly at $\infty$. Then $(X,\infty,\mathbf{P}\breve{W})$ is a cone
situation satisfying none of (i), (ii), nor (iii).

The following illustrates how the hypotheses of 2.1 translate to geometric
consequences on $X$ and its Fano scheme $S$ of lines; the first statement
gives sufficient conditions—although not necessary, as shown by 2.2(vi)—for
when $S$ is of expected dimension $2$, whereas the second classifies cone
situations satisfying the two smoothness hypotheses on
$(X,\infty,\mathbf{P}\breve{W})$.

######  2.3 Lemma. —

If the cone situation $(X,\infty,\mathbf{P}\breve{W})$ furthermore satisfies

  1. (i)

conditions 2.1(i) and 2.1(ii), then $\dim S=2$;

  2. (ii)

conditions 2.1(ii) and 2.1(iii), then it is projectively equivalent to either
2.2(i) or 2.2(ii).

###### Proof.

Suppose that $(X,\infty,\mathbf{P}\breve{W})$ satisfies 2.1(i) and 2.1(ii).
Then $X$ is not a cone: otherwise, its vertex $x$, which is different from the
smooth point $\infty$, would span a plane $\langle x,\ell\rangle$ through
$\infty$ with any line $\ell\subset X\cap\mathbf{P}\breve{W}$ through $\infty$
and not passing through $x$. The comments in 1.10 show that
$\dim\operatorname{Sing}X\leq 1$. Suppose $x\in X$ is a singular point with
underlying linear space $K\subset V$. Lines in $X$ through $x$ are contained
in

$X\cap\mathbf{P}K^{[1],\perp}\cap\mathbf{P}K^{\perp,[-1]}=X\cap\mathbf{T}_{X,x}\cap\mathbf{P}K^{\perp,[-1]}=X\cap\mathbf{P}K^{\perp,[-1]}$

since the underlying linear space pairs to $0$ with $K$ on either side of
$\beta$: see also [Che23b, 3.1]. Since $x$ is not a vertex, this is a surface
which is a cone with vertex $x$ over a $q$-bic curve. Thus the locus of lines
through $x$ is of dimension $1$. Therefore discussion of 1.8 implies that
$\dim S=2$.

Suppose now that $(X,\infty,\mathbf{P}\breve{W})$ furthermore satisfies
2.1(iii). If $X$ itself is smooth, then it must be 2.2(i), so suppose $X$ is
singular. Since $\infty$ is a smooth point, and since
$\mathbf{P}\breve{W}=\mathbf{T}_{X,\infty}$ intersects $X$ at a cone with a
smooth base, the singular locus of $X$ must be a single point $x=\mathbf{P}K$
disjoint from $\mathbf{P}\breve{W}$. But $W=L^{[1],\perp}$, so this means that
the natural map $K\to L^{[1],\vee}$ is nonzero, whence an isomorphism.
Splitting $V=\breve{W}\oplus K$, it now follows that

$V^{[1],\perp}=\ker(\breve{W}\oplus
K\to\breve{W}^{[1],\vee})=\ker(\breve{W}\to\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{W}^{[1],\vee})=L.$

Thus the restriction of $\beta$ to $U\coloneqq L\oplus K$ is of type
$\mathbf{N}_{2}$. Since $U^{[1],\perp}=L^{[1],\perp}$ and
$U^{\perp}=K^{\perp}$ are distinct hyperplanes, their intersection in $V$
gives the orthogonal complement to $U$. Whence $X$ is of type
$\mathbf{1}^{\oplus 3}\oplus\mathbf{N}_{2}$ and $\infty$ is the smooth point
of the subform $\mathbf{N}_{2}$. ∎

Let $(X,\infty,\mathbf{P}\breve{W})$ be a cone situation over the $q$-bic
curve $C$ and let $S$ be the Fano scheme of lines of $X$. Then there is a
canonical closed immersion $C\hookrightarrow S$ identifying it with the closed
subscheme

$C_{\infty,\mathbf{P}\breve{W}}\coloneqq\set{[\ell]\in
S:\infty\in\ell\subset\mathbf{P}\breve{W}}$

of lines containing $\infty$ and contained in $X\cap\mathbf{P}\breve{W}$, so
that the Plücker polarization $\mathcal{O}_{S}(1)$ pulls back to the planar
polarization $\mathcal{O}_{C}(1)$. The hypotheses of 2.1 affect
$C_{\infty,\mathbf{P}\breve{W}}$ as follows:

######  2.4 Lemma. —

Suppose the cone situation $(X,\infty,\mathbf{P}\breve{W})$ furthermore
satisfies

  1. (i)

condition 2.1(i), then the closed subsets $\set{[\ell]\in S:\infty\in\ell}$
and $\set{[\ell]\in S:\ell\subset\mathbf{P}\breve{W}}$ coincide;

  2. (ii)

condition 2.1(ii), then $C_{\infty,\mathbf{P}\breve{W}}$ coincides with the
subscheme $C_{\infty}$ of lines in $X$ through $\infty$; and

  3. (iii)

conditions 2.1(ii) and 2.1(iii), then
$\mathcal{N}_{C_{\infty,\mathbf{P}\breve{W}}/S}\cong\mathcal{O}_{C}(-q+1)$.

###### Proof.

Establish (i) through its contrapositive: To see “$\supseteq$”, if there were
a line $\ell\subset X\cap\mathbf{P}\breve{W}$ not containing $\infty$, then
the cone $X\cap\mathbf{P}\breve{W}$ would contain the plane
$\langle\infty,\ell\rangle$ spanned by $\infty$ and $\ell$. To see
“$\subseteq$”, if there were a line $\ell\ni\infty$ not contained in
$\mathbf{P}\breve{W}$, then the orthogonals of $L$ contain not only
$\breve{W}$, by the comments of 2.1, but also the subspace underlying $\ell$.
Therefore $L$ lies in the radical of $\beta$, so $X$ is a cone over a $q$-bic
surface with vertex containing $\infty$. The result now follows since every
$q$-bic surface contains a line by 1.9.

For (ii), the comments following 2.1 imply that $\mathbf{P}\breve{W}$ is the
embedded tangent space of $X$ at $\infty$. Since any line in $X$ through
$\infty$ is necessarily contained in $\mathbf{T}_{X,\infty}$, the moduli
problem underlying $C_{\infty}$ is a closed subfunctor of that of
$C_{\infty,\mathbf{P}\breve{W}}$. Since the reverse inclusion is clear,
equality holds.

For (iii), observe that there is a short exact sequence of locally free
sheaves

$0\to\mathcal{T}_{C}\to\mathcal{T}_{S}\rvert_{C}\to\mathcal{N}_{C/S}\to 0$

where $C_{\infty,\mathbf{P}\breve{W}}$ is identified with $C$: indeed, this is
because the assumptions together with the discussion in 1.8 imply that $S$ is
smooth along $C$. Taking determinants then gives the normal bundle. ∎

The cone situation gives a way of transforming lines in $X$ to points of $C$,
inducing a rational map $S\dashrightarrow C$ in good cases. Some hypothesis on
$(X,\infty,\mathbf{P}\breve{W})$ is necessary to exclude, for instance,
example 2.2(vi), wherein the locus of lines through $\infty$ forms an
irreducible component of $S$. In the following, write
$\operatorname{proj}_{\infty}\colon\mathbf{P}V\dashrightarrow\mathbf{P}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{V}$
for the linear projection away from $\infty$, and let $S^{\circ}\coloneqq
S\setminus C_{\infty,\mathbf{P}\breve{W}}$.

######  2.5 Lemma. —

If the cone situation $(X,\infty,\mathbf{P}\breve{W})$ satisfies 2.1(i), then
there is a rational map $\varphi\colon S\dashrightarrow C$ given on points
$[\ell]\in S^{\circ}$ by

$\varphi([\ell])=\operatorname{proj}_{\infty}(\ell\cap\mathbf{P}\breve{W})=\operatorname{proj}_{\infty}(\ell)\cap\mathbf{P}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{W}.$

###### Proof.

The point is that 2.4(i) implies that the indeterminacy locus of the formula
is a proper closed subset not containing any irreducible components of $S$. ∎

### 2.6.

— Towards a resolution of $\varphi$, consider the scheme parameterizing
triples

$\mathbf{P}\coloneqq\Set{(y\mapsto
y_{0}\in\ell_{0}):y\in\mathbf{P}\breve{W},[\ell_{0}]\in\mathbf{G}(2,\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{V}),y_{0}\in C}$

consisting of a point $y\in\mathbf{P}\breve{W}$, a line
$\ell_{0}\subset\mathbf{P}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{V}$,
and a point $y_{0}\in\ell_{0}\cap C$, such that $y$ is contained in the line
in $\mathbf{P}\breve{W}$ determined by $y_{0}$. Equivalently, $\mathbf{P}$ is
the product $\mathbf{P}\mathcal{V}_{1}\times_{C}\mathbf{P}\mathcal{V}_{2}$ of
projective bundles over $C$, where $\mathcal{V}_{1}$ and $\mathcal{V}_{2}$ are
defined via pullback of the left square and pushout of the right square,
respectively, of the following commutative diagram:

${0}$${\mathcal{V}_{1}}$${V_{C}}$${\mathcal{V}_{2}}$${0}$${\breve{W}_{C}}$${\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{V}_{C}}$${0}$${\mathcal{O}_{C}(-1)}$${\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{W}_{C}}$${\mathcal{T}}$${0}$

where $\mathcal{T}\coloneqq\mathcal{T}_{\mathbf{P}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{W}}(-1)\rvert_{C}$, $V_{C}\coloneqq V\otimes\mathcal{O}_{C}$, and
similarly for the others. In particular, there are split short exact sequences

$0\to L_{C}\to\mathcal{V}_{1}\to\mathcal{O}_{C}(-1)\to 0\quad\text{and}\quad
0\to\mathcal{T}\to\mathcal{V}_{2}\to(V/\breve{W})_{C}\to 0.$

Let $\mathbf{P}^{\circ}$ be the open subscheme of $\mathbf{P}$ where
$y\neq\infty$ and $\ell_{0}\not\subset\mathbf{P}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{W}$.
Its closed complement is the union of two effective Cartier divisor, with
points

$\mathbf{P}\setminus\mathbf{P}^{\circ}=\Set{(y\mapsto
y_{0}\in\ell_{0}):y=\infty\;\text{or}\;\ell_{0}\subset\mathbf{P}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{W}}=\mathbf{P}\mathcal{V}_{1}\times_{C}\mathbf{P}\mathcal{T}\cup\mathbf{P}L_{C}\times_{C}\mathbf{P}\mathcal{V}_{2}.$

An equation of $\mathbf{P}\setminus\mathbf{P}^{\circ}$ is as follows: Let
$\pi_{i}\colon\mathbf{P}\mathcal{V}_{i}\to C$ and $\pi\colon\mathbf{P}\to C$
be the structure morphisms, let
$\operatorname{pr}_{i}\colon\mathbf{P}\to\mathbf{P}\mathcal{V}_{i}$ be the
projections, and for any $\mathcal{O}_{\mathbf{P}}$-module $\mathcal{F}$ and
$a,b\in\mathbf{Z}$, write

$\mathcal{F}(a,b)\coloneqq\mathcal{F}\otimes\operatorname{pr}_{1}^{*}\mathcal{O}_{\pi_{1}}(a)\otimes\operatorname{pr}_{2}^{*}\mathcal{O}_{\pi_{2}}(b).$

Then $\mathbf{P}\setminus\mathbf{P}^{\circ}=\mathrm{V}(u_{1}u_{2})$, where
$u_{i}$ is the map obtained by composing the Euler section
$\mathrm{eu}_{\pi_{i}}$ with the quotient map for $\mathcal{V}_{i}$ in the two
sequences above:

$u_{1}\colon\mathcal{O}_{\mathbf{P}}(-1,0)\hookrightarrow\pi^{*}\mathcal{V}_{1}\twoheadrightarrow\pi^{*}\mathcal{O}_{C}(-1)\quad\text{and}\quad
u_{2}\colon\mathcal{O}_{\mathbf{P}}(0,-1)\hookrightarrow\pi^{*}\mathcal{V}_{2}\twoheadrightarrow(V/\breve{W})_{\mathbf{P}}.$

Returning to $\varphi$, observe that the morphism of 2.5 factors through a
morphism $S^{\circ}\to\mathbf{P}^{\circ}$ given by

$[\ell]\mapsto\big{(}\ell\cap\mathbf{P}\breve{W}\mapsto\varphi([\ell])\in\operatorname{proj}_{\infty}(\ell)\big{)}.$

The image of this morphism can be described in terms of the geometry of
$X_{\ell_{0}}\coloneqq X\cap P_{\ell_{0}}$, where
$P_{\ell_{0}}\coloneqq\operatorname{proj}_{\infty}^{-1}(\ell_{0})$ is the
plane in $\mathbf{P}V$ spanned by $\infty$ and a line
$\ell_{0}\subset\mathbf{P}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{V}$:

######  2.7 Lemma. —

The image of $S^{\circ}\to\mathbf{P}^{\circ}$ is the scheme

$T^{\circ}=\Set{(y\mapsto
y_{0}\in\ell_{0})\in\mathbf{P}^{\circ}:X_{\ell_{0}}\;\text{is a cone with
vertex}\;y}.$

If $(X,\infty,\mathbf{P}\breve{W})$ satisfies 2.1(i), then the morphism
$S^{\circ}\to T^{\circ}$ is quasi-finite of degree $q$.

###### Proof.

A point $(y\mapsto y_{0}\in\ell_{0})\in\mathbf{P}^{\circ}$ lies in the image
$T^{\circ}$ if and only if there is a line $\ell$ contained in
$X_{\ell_{0}}\setminus\infty$ through $y$. If $P_{\ell_{0}}\subset X$, then
any $y$ is a vertex of the plane $X_{\ell_{0}}$, and any line in
$X_{\ell_{0}}$ through $y$ and not passing through $\infty$ witnesses
membership in $T^{\circ}$. Otherwise, $X_{\ell_{0}}$ is a $q$-bic curve which
contains the line

$\langle
y,\infty\rangle=\operatorname{proj}_{\infty}^{-1}(y_{0})=\operatorname{proj}_{\infty}^{-1}(\ell_{0}\cap\mathbf{P}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{W})=P_{\ell_{0}}\cap\mathbf{P}\breve{W}$

spanned by $y$ and $\infty$ as an irreducible component. Thus $(y\mapsto
y_{0}\in\ell_{0})$ is contained in $T^{\circ}$ if and only if the residual
curve $X_{\ell_{0}}-\ell_{y,\infty}$ contains a line passing through $y$.
Classification of $q$-bic curves, as in 1.4, shows that this happens if and
only if $X_{\ell_{0}}$ is a cone with vertex $y$. Finally, if the cone
situation satisfies 2.1(i), then only this second case occurs, and this
analysis shows that the fibres of $S^{\circ}\to T^{\circ}$ are the length $q$
schemes parameterizing the lines in $X_{\ell_{0}}-\ell_{y,\infty}$. ∎

###  2.8. Closure of $T^{\circ}$

— To describe the closure of $T^{\circ}$ in $\mathbf{P}$, first extend the
moduli description in 2.7 to all of $\mathbf{P}$ by considering the locally
free $\mathcal{O}_{\mathbf{P}}$-module $\mathcal{P}$ of rank $3$ defined by
the diagram

${0}$${\pi^{*}\mathcal{V}_{1}}$${V_{\mathbf{P}}}$${\pi^{*}\mathcal{V}_{2}}$${0}$${0}$${\pi^{*}\mathcal{V}_{1}}$${\mathcal{P}}$${\mathcal{O}_{\mathbf{P}}(0,-1)}$${0}$$\scriptstyle{\mathrm{eu}_{\pi_{2}}}$

where the upper sequence is as in 2.6. The fibre at a point $(y\mapsto
y_{0}\in\ell_{0})\in\mathbf{P}$ of

  * –

$\mathcal{P}$ is the subspace of $V$ underlying the plane $P_{\ell_{0}}$;

  * –

$\pi^{*}\mathcal{V}_{1}$ is the line $\ell_{0,\infty}\coloneqq\langle
y_{0},\infty\rangle$; and

  * –

the tautological subbundle
$\mathcal{O}_{\mathbf{P}}(-1,0)\hookrightarrow\pi^{*}\mathcal{V}_{1}$ is the
point $y$.

Let
$\beta_{\mathcal{P}}\colon\mathcal{P}^{[1]}\otimes\mathcal{P}\to\mathcal{O}_{\mathbf{P}}$
be the restriction of the $q$-bic form $\beta$. Since $\mathcal{V}_{1}$ is
isotropic—it parameterizes lines in $X$!—the adjoints of $\beta_{\mathcal{P}}$
induce maps

$\beta^{\vee}_{\mathcal{P}}\colon\pi^{*}\mathcal{V}_{1}^{[1]}\xrightarrow{\beta^{\vee}\rvert_{\mathcal{V}_{1}}}\pi^{*}\mathcal{V}_{2}^{\vee}\xrightarrow{\mathrm{eu}_{\pi_{2}}^{\vee}}\mathcal{O}_{\mathbf{P}}(0,1)\quad\text{and}\quad\beta_{\mathcal{P}}\colon\pi^{*}\mathcal{V}_{1}\xrightarrow{\beta\rvert_{\mathcal{V}_{1}}}\pi^{*}\mathcal{V}_{2}^{[1],\vee}\xrightarrow{\mathrm{eu}_{\pi_{2}}^{[1],\vee}}\mathcal{O}_{\mathbf{P}}(0,q).$

Set $v_{1}\coloneqq\beta^{\vee}_{\mathcal{P}}\circ\mathrm{eu}_{\pi_{1}}^{[1]}$
and $v_{2}\coloneqq\beta_{\mathcal{P}}\circ\mathrm{eu}_{\pi_{1}}$. Then
$v\coloneqq(v_{1},v_{2})\colon\mathcal{O}_{\mathbf{P}}\to\mathcal{O}_{\mathbf{P}}(q,1)\oplus\mathcal{O}_{\mathbf{P}}(1,q)$
vanishes at points where the line in $\mathcal{P}$ given by
$\mathrm{eu}_{\pi_{1}}$ lies in the radical of $\beta_{\mathcal{P}}$, or,
geometrically:

$T^{\prime}\coloneqq\mathrm{V}(v)=\Set{(y\mapsto
y_{0}\in\ell_{0})\in\mathbf{P}:X_{\ell_{0}}\;\text{is a cone with
vertex}\;y}.$

In particular, $T^{\circ}=T^{\prime}\cap\mathbf{P}^{\circ}$.

This contains too much: for instance, $T^{\prime}$ contains the intersection
of the irreducible components of $\mathbf{P}\setminus\mathbf{P}^{\circ}$,
consisting of $(\infty\mapsto y_{0}\in\ell_{0})$ where
$\ell_{0}\subset\mathbf{P}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{W}$.
In fact, the map $v=(v_{1},v_{2})^{\vee}$ often factors through
$u\coloneqq(u_{1},u_{2})^{\vee}$. To explain, given splittings
$\mathcal{V}_{1}\cong\mathcal{O}_{C}(-1)\oplus L_{C}$ and
$\mathcal{V}_{2}\cong\mathcal{T}\oplus(V/\breve{W})_{C}$, write

$u_{1}^{\prime}\colon\mathcal{O}_{\mathbf{P}}(-1,0)\stackrel{{\scriptstyle\mathrm{eu}_{\pi_{1}}}}{{\longrightarrow}}\pi^{*}\mathcal{V}_{1}\longrightarrow
L_{\mathbf{P}}\quad\text{and}\quad
u_{2}^{\prime}\colon\mathcal{O}_{\mathbf{P}}(0,-1)\stackrel{{\scriptstyle\mathrm{eu}_{\pi_{2}}}}{{\longrightarrow}}\pi^{*}\mathcal{V}_{2}\longrightarrow\pi^{*}\mathcal{T}$

for the projection of the Euler sections to the subbundle.

######  2.9 Lemma. —

Assume there exists a $2$-dimensional subspace $U\subset V$ such that

  1. (i)

$U\cap\breve{W}=L$, and

  2. (ii)

$U$ admits an orthogonal complement $W$ in $V$.

Then the induced splittings $\mathcal{V}_{1}\cong\mathcal{O}_{C}(-1)\oplus
L_{C}$ and $\mathcal{V}_{2}\cong\mathcal{T}\oplus(V/\breve{W})_{C}$ are such
that the adjoint maps

$\displaystyle\beta^{\vee}\rvert_{\mathcal{V}_{1}}$
$\displaystyle\eqqcolon\beta^{\vee}_{1}\oplus\beta^{\vee}_{2}\colon\mathcal{O}_{C}(-q)\oplus
L_{C}^{[1]}\to\mathcal{T}^{\vee}\oplus(V/\breve{W})_{C}^{\vee},$
$\displaystyle\beta\rvert_{\mathcal{V}_{1}}$
$\displaystyle\eqqcolon\beta_{1}^{\phantom{\vee}}\oplus\beta_{2}^{\phantom{\vee}}\colon\mathcal{O}_{C}(-1)\oplus
L_{C}^{\phantom{[1]}}\to\mathcal{T}^{[1],\vee}\oplus(V/\breve{W})_{C}^{[1],\vee}$

are diagonal, and there exists a factorization
$v=v^{\prime}\circ(u_{1},u_{2})^{\vee}$ where $v^{\prime}$ is the map

$\begin{pmatrix}u_{2}^{\prime}\cdot\beta_{1}^{\vee}\cdot
u_{1}^{q-1}&\beta_{2}^{\vee}\cdot u_{1}^{\prime q}\\\ u_{2}^{\prime
q}\cdot\beta_{1}&u_{2}^{q-1}\cdot\beta_{2}\cdot
u_{1}^{\prime}\end{pmatrix}\colon\mathcal{O}_{\mathbf{P}}(1,0)\otimes\pi^{*}\mathcal{O}_{C}(-1)\oplus\mathcal{O}_{\mathbf{P}}(0,1)\otimes(V/\breve{W})_{\mathbf{P}}\to\mathcal{O}_{\mathbf{P}}(q,1)\oplus\mathcal{O}_{\mathbf{P}}(1,q).$

###### Proof.

The decomposition $V=W\oplus U$ induces splittings
$\mathcal{V}_{1}\cong\mathcal{O}_{C}(-1)\oplus L_{C}$ and
$\mathcal{V}_{2}\cong\mathcal{T}\oplus(V/\breve{W})_{C}$ by intersecting
$\mathcal{V}_{1}$ with $W_{C}$ as subbundles of $V_{C}$, and by projecting
$U_{C}$ to $\mathcal{V}_{2}$, respectively. The adjoint maps of $\beta$ are
now diagonal since $W$ and $U$ are orthogonal complements. The Euler sections
split as $\mathrm{eu}_{\pi_{1}}=(u_{1},u_{1}^{\prime})^{\vee}$ and
$\mathrm{eu}_{\pi_{2}}=(u_{2}^{\prime},u_{2})^{\vee}$, and so

$\displaystyle v_{1}$
$\displaystyle=\mathrm{eu}_{\pi_{2}}^{\vee}\cdot\beta^{\vee}\rvert_{\mathcal{V}_{1}}\cdot\mathrm{eu}_{\pi_{1}}^{[1]}=u_{2}^{\prime}\cdot\beta_{1}^{\vee}\cdot
u_{1}^{q}+u_{2}\cdot\beta_{2}^{\vee}\cdot u_{1}^{\prime q},\;\text{and}$
$\displaystyle v_{2}$
$\displaystyle=\mathrm{eu}_{\pi_{2}}^{[1],\vee}\cdot\beta\rvert_{\mathcal{V}_{1}}\cdot\mathrm{eu}_{\pi_{1}}=u_{2}^{\prime
q}\cdot\beta_{1}\cdot u_{1}+u_{2}^{q}\cdot\beta_{2}\cdot u_{1}^{\prime}.$

Since $u_{1}$ and $u_{2}$ are locally multiplication by a scalar, their action
commutes with the other maps present, and a direct computation shows that
there is a factorization $v=v\circ(u_{1},u_{2})^{\vee}$, as claimed. ∎

The hypotheses of 2.9 are satisfied, for example, in all of the cone
situations 2.2(i)–(iii), so, in particular, in all smooth cone situations as
in 2.3(ii): in 2.2(i), $U$ may be taken to be the span of $L$ with any other
isotropic Hermitian vector not lying in $\breve{W}$; in 2.2(ii) and 2.2(iii),
$U$ is necessarily the subspace supporting the subform of type
$\mathbf{N}_{2}$. In contrast, no $U$ exists in 2.2(v).

This setting provides a candidate for the Zariski closure of $T^{\circ}$ in
$\mathbf{P}$. To state the result, write

$\displaystyle\mathcal{E}_{2}$
$\displaystyle\coloneqq\mathcal{O}_{\mathbf{P}}(1,0)\otimes\pi^{*}\mathcal{O}_{C}(-1)\oplus\mathcal{O}_{\mathbf{P}}(0,1)\otimes(V/\breve{W})_{\mathbf{P}},$
$\displaystyle\mathcal{E}_{1}$
$\displaystyle\coloneqq\mathcal{O}_{\mathbf{P}}(q,1)\oplus\mathcal{O}_{\mathbf{P}}(1,q)\oplus\det(\mathcal{E}_{2}),$

and $\wedge u\colon\mathcal{E}_{2}\to\det(E_{2})$ for the natural surjection
to the torsion-free quotient of
$\operatorname{coker}(u\colon\mathcal{O}_{\mathbf{P}}\to\mathcal{E}_{2})$.

######  2.10 Proposition. —

In the setting of 2.9, the scheme $T\coloneqq
T^{\prime}\cap\mathrm{V}(\det(v^{\prime}))$ contains the Zariski closure of
$T^{\circ}$ in $\mathbf{P}$, and is the rank $1$ degeneracy locus of

$\phi\coloneqq\left(\begin{smallmatrix}v^{\prime}\\\ \wedge
u\end{smallmatrix}\right)\colon\mathcal{E}_{2}\to\mathcal{E}_{1}$

If furthermore $\dim S^{\circ}=2$, then $\dim T=2$, $T$ is connected,
Cohen–Macaulay, and there is an exact complex of sheaves on $\mathbf{P}$ given
by

$0\longrightarrow\mathcal{E}_{2}(-q-1,-q-1)\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}\mathcal{E}_{1}(-q-1,-q-1)\stackrel{{\scriptstyle\wedge^{2}\phi^{\vee}}}{{\longrightarrow}}\mathcal{O}_{\mathbf{P}}\longrightarrow\mathcal{O}_{T}\longrightarrow
0.$

###### Proof.

By 2.8, $T^{\circ}$ is the vanishing locus of $v=v^{\prime}\circ u$ on the
open subscheme $\mathbf{P}^{\circ}$ where neither $u_{1}$ nor $u_{2}$ vanish.
Therefore $v^{\prime}$ has rank at most $1$ on $T^{\circ}$, meaning
$\det(v^{\prime})\rvert_{T^{\circ}}=0$ and $T^{\circ}\subseteq T$. To express
$T$ as a degeneracy locus, restrict first to the vanishing locus of
$\det(v^{\prime})$, which is but the rank $1$ locus of
$v^{\prime}\colon\mathcal{E}_{2}\to\mathcal{O}_{\mathbf{P}}(q,1)\oplus\mathcal{O}_{\mathbf{P}}(1,q)$.
Now the factorization $v=v^{\prime}\circ u$ means that $T$ is the locus where
$\operatorname{im}(u)\subseteq\ker(v^{\prime})$. On the one hand, $\ker(\wedge
u)$ is the saturation in $\mathcal{E}_{2}$ of $\operatorname{im}(u)$. On the
other hand, $v^{\prime}$ is a map between locally free sheaves, and so
$\ker(v^{\prime})$ is saturated. Therefore $T$ is equivalently the locus in
$\mathrm{V}(\det(v^{\prime}))$ where $\ker(\wedge
u)\subseteq\ker(v^{\prime})$. Altogether, this means that $T$ is the locus in
$\mathbf{P}$ where $\phi\coloneqq\left(\begin{smallmatrix}v^{\prime}\\\ \wedge
u\end{smallmatrix}\right)$ has rank at most $1$, as required.

For the remainder, note that its description as a degeneracy locus implies
$\dim T\leq 2$. Suppose that $\dim S^{\circ}=2$. Then its image $T^{\circ}$
has dimension at most $2$. Since $T\setminus
T^{\circ}\subseteq\mathbf{P}\setminus\mathbf{P}^{\circ}$, it follows that
$\dim T=2$. Since the base curve $C$ is reduced, it is Cohen–Macaulay, and
thus so is $\mathbf{P}$. With this, the remaining properties follow from the
fact that $T$ is a degeneracy locus of expected dimension: see [HE70, Theorem
1], or also [Ful98, Theorem 14.3(c)], for Cohen–Macaulayness; see [Laz04,
Theorem B.2.2(ii)] for exactness of the Eagon–Northcott complex
$(\mathrm{EN}_{0})$ associated with $\phi$. Finally, connectedness of $T$
follows from that of $S$, see 1.8. ∎

In many cases, $T$ is the closure of $T^{\circ}$. This is verified for smooth
cone situations in 3.2.

### 2.11.

— Write $\mathcal{T}_{\pi_{i}}$ for the pullback to $\mathbf{P}$ of the
relative tangent bundle of $\pi_{i}\colon\mathbf{P}\mathcal{V}_{i}\to C$, and
set

$\mathcal{V}\coloneqq\mathcal{H}\big{(}\mathcal{O}_{\mathbf{P}}(-1,0)\hookrightarrow
V_{\mathbf{P}}\twoheadrightarrow\mathcal{T}_{\pi_{2}}(0,-1)\big{)}\cong\operatorname{coker}\big{(}\mathcal{O}_{\mathbf{P}}(-1,0)\hookrightarrow\pi^{*}\mathcal{V}_{1}\hookrightarrow\mathcal{P}\big{)}$

where $\mathcal{H}$ extracts the homology sheaf, and $\mathcal{P}$ is as in
2.8. Its associated $\mathbf{P}^{1}$-bundle is

$\mathbf{P}\mathcal{V}=\Set{((y\in\ell)\mapsto(y_{0}\in\ell_{0})):(y\mapsto
y_{0}\in\ell_{0})\in\mathbf{P}\;\text{and}\;\ell\;\text{a line
in}\;P_{\ell_{0}}}.$

Extracting the line $[\ell]$ yields a birational morphism
$\mathbf{P}\mathcal{V}\to\mathbf{G}$ which is an isomorphism away from the
locus where either $\infty\in\ell$ or $\ell\subset\mathbf{P}\breve{W}$. The
discussion preceding 2.7 implies that $S$ is contained in the image of
$\mathbf{P}\mathcal{V}$ and is not completely contained in the non-isomorphism
locus, so it has a well-defined strict transform $\tilde{S}$ along
$\mathbf{P}\mathcal{V}\to\mathbf{G}$ that has a morphism $\tilde{S}\to T$;
when $(X,\infty,\mathbf{P}\breve{W})$ satisfies 2.1(i), this resolves the
rational map $\varphi\colon S\dashrightarrow T$ from 2.5. In summary, there is
a commutative diagram:

${S}$${\tilde{S}}$${\mathbf{P}\mathcal{V}}$${T}$${\mathbf{P}}$${C\makebox[0.0pt][l]{\,.}}$$\scriptstyle{\pi}$

From now on, assume that $(X,\infty,\mathbf{P}\breve{W})$ satisfies 2.1(i).
The remainder of this section is dedicated to constructing equations for
$\tilde{S}$ by describing it as a bundle of $q$-bic points over $T$. Write
$\rho\colon\mathbf{P}\mathcal{V}_{T}\to T$ for the projection of
$\mathbf{P}\mathcal{V}\to\mathbf{P}$ restricted to $T$. Then 2.7 implies that
$\tilde{S}$ is a hypersurface in $\mathbf{P}\mathcal{V}_{T}$. Let
$(\mathcal{P}_{T},\beta_{\mathcal{P}_{T}})$ be the restriction to $T$ of the
$q$-bic form $(\mathcal{P},\beta_{\mathcal{P}})$ from 2.8. As a first step:

######  2.12 Lemma. —

The $q$-bic form $\beta_{\mathcal{P}_{T}}$ induces a $q$-bic form
$\beta_{\mathcal{V}_{T}}\colon\mathcal{V}_{T}^{[1]}\otimes\mathcal{V}_{T}\to\mathcal{O}_{T}$
whose $q$-bic equation

$\beta_{\mathcal{V}_{T}}(\mathrm{eu}_{\rho}^{[1]},\mathrm{eu}_{\rho})\colon\mathcal{O}_{\rho}(-q-1)\hookrightarrow\rho^{*}\mathcal{V}_{T}^{[1]}\otimes\rho^{*}\mathcal{V}_{T}\rightarrow\mathcal{O}_{\mathbf{P}\mathcal{V}_{T}},$

vanishes at
$((y\in\ell)\mapsto(y_{0}\in\ell_{0}))\in\mathbf{P}\mathcal{V}_{T}$ if and
only if $\ell$ is an isotropic line for $\beta$. In particular, this section
vanishes on the strict transform $\tilde{S}$ of $S$.

###### Proof.

By 2.8, the intersection $X_{\ell_{0}}=X\cap P_{\ell_{0}}$ is a cone with
vertex $y$ for every $(y\mapsto y_{0}\in\ell_{0})\in T$. The description of
the fibres of $\mathcal{P}_{T}$ from 2.8 implies that the tautological
subbundle $\mathcal{O}_{T}(-1,0)\hookrightarrow\mathcal{P}_{T}$ lies in the
radical of the form $\beta_{\mathcal{P}_{T}}$, and so it induces a $q$-bic
form $\beta_{\mathcal{V}_{T}}$ on the quotient $\mathcal{V}_{T}$. Comparing
again with the description of the fibres of $\mathcal{P}_{T}$ shows that the
fibre of the tautological subbundle $\mathcal{O}_{\rho}(-1)$ at
$((y\in\ell)\mapsto(y_{0}\in\ell_{0}))$ extracts the subspace of $V$
underlying $\ell$, whence the latter statement. ∎

### 2.13.

— Comparing the homology sheaf construction of $\mathcal{V}$ from 2.11 with
the sequences for $\mathcal{V}_{1}$ and $\mathcal{V}_{2}$ from 2.6 gives a
canonical short exact sequence

$0\to\mathcal{T}_{\pi_{1}}(-1,0)\to\mathcal{V}\to\mathcal{O}_{\mathbf{P}}(0,-1)\to
0.$

The subbundle may be identified via the Euler sequence for
$\mathbf{P}\mathcal{V}_{1}\to C$ as:

$\mathcal{T}_{\pi_{1}}(-1,0)=\operatorname{coker}(\mathrm{eu}_{\pi_{1}}\colon\mathcal{O}_{\mathbf{P}}(-1,0)\to\pi^{*}\mathcal{V}_{1})\cong\det(\pi^{*}\mathcal{V}_{1})(1,0)\cong\mathcal{O}_{\mathbf{P}}(1,0)\otimes\pi^{*}\mathcal{O}_{C}(-1)\otimes
L.$

Its inverse image under the quotient map $\mathcal{P}\to\mathcal{V}$ is the
subbundle $\pi^{*}\mathcal{V}_{1}\subset\mathcal{P}$, so its points are

$\mathbf{P}(\mathcal{T}_{\pi_{1}}(-1,0))=\Set{((y\in\ell)\mapsto(y_{0}\in\ell_{0}))\in\mathbf{P}\mathcal{V}:\ell=\langle
y_{0},\infty\rangle}.$

Since $\ell=\langle y_{0},\infty\rangle\subset\mathbf{P}\breve{W}$ whenever
$(y\mapsto y_{0}\in\ell_{0})\in T$, this subbundle is isotropic for
$\beta_{\mathcal{V}_{T}}$ by 2.12, yielding the following observation:

######  2.14 Lemma. —

The $q$-bic equation
$\beta_{\mathcal{V}_{T}}(\mathrm{eu}_{\rho}^{[1]},\mathrm{eu}_{\rho})$ from
2.12 vanishes on $\mathbf{P}(\mathcal{T}_{\pi_{1}}(-1,0)\rvert_{T})$, and so
it factors through the section

$v_{3}\coloneqq
u_{3}^{-1}\beta_{\mathcal{V}_{T}}(\mathrm{eu}^{[1]}_{\rho},\mathrm{eu}_{\rho})\colon\mathcal{O}_{\mathbf{P}\mathcal{V}_{T}}\to\mathcal{O}_{\rho}(q)\otimes\rho^{*}\mathcal{O}_{T}(0,1)$

where
$u_{3}\colon\mathcal{O}_{\rho}(-1)\to\rho^{*}\mathcal{V}_{T}\to\rho^{*}\mathcal{O}_{T}(0,-1)$
is the equation of the subbundle. ∎

######  2.15 Lemma. —

The section $v_{3}$ vanishes on $\tilde{S}$, and $\tilde{S}\to T$ is finite
flat of degree $q$ onto its image.

###### Proof.

The discussion of 2.11 and 2.14 together with 2.4(i) implies that the
intersection of $\tilde{S}$ with the exceptional locus of
$\mathbf{P}\mathcal{V}\to\mathbf{G}$ is contained in
$\mathbf{P}(\mathcal{T}_{\pi_{1}}(-1,0)\rvert_{T})$. Therefore $v_{3}$
vanishes on $\tilde{S}$ if and only if it vanishes on
$S^{\circ}\cong\tilde{S}\setminus\mathbf{P}(\mathcal{T}_{\pi_{1}}(-1,0)\rvert_{T})$.
Since $u_{3}$ is invertible on the latter open subscheme, $v_{3}$ vanishes on
$S^{\circ}$ by 2.12.

Since $S^{\circ}\to T$ is quasi-finite of degree $q$ by 2.7, the final
statement follows upon showing that $\mathrm{V}(v_{3})\to T$ is finite flat of
degree $q$. Since $v_{3}$ is degree $q$ on each fibre of
$\mathbf{P}\mathcal{V}_{T}\to T$ by 2.14, it suffices to see that $v_{3}$ does
not vanish on an entire fibre. But if $v_{3}$ did vanish on the fibre over
$(y\mapsto y_{0}\in\ell_{0})\in T$, 2.12 would imply that all lines
$\ell\subset P_{\ell_{0}}$ passing through $y$ are isotropic for $\beta$, and
hence $P_{\ell_{0}}$ would be contained in $X$. This is impossible with
condition 2.1(i). ∎

##  3\. Smooth cone situation

The most useful cone situations that arise when studying $q$-bic threefolds
which are either smooth or of type $\mathbf{1}^{\oplus 3}\oplus\mathbf{N}_{2}$
are the _smooth cone situations_ : those $(X,\infty,\mathbf{P}\breve{W})$
which satisfy 2.1(ii) and 2.1(iii), and which were classified in 2.3(ii) to be
as in either example 2.2(i) or 2.2(ii). In this setting, the scheme $T$
constructed in 2.10 is the Zariski closure of $T^{\circ}$ and also a quotient
of $\tilde{S}$ by a finite group scheme of order $q$: see 3.2 and 3.5.
Furthermore, $\tilde{S}$ is a blowup of $S$ along smooth points, see 3.7; this
implies that $\varphi$ from 2.5 extends to a morphism $S\to C$ and that the
sheaf $\mathbf{R}^{1}\varphi_{*}\mathcal{O}_{S}$ carries a canonical
filtration, crucial for the computations that follow: see 3.9 and 3.10.

Throughout, let $(X,\infty,\mathbf{P}\breve{W})$ be a smooth cone situation,
fix an orthogonal decomposition $V\cong W\oplus U$ as in 2.9, and write
$\mathcal{V}_{1}\cong\mathcal{O}_{C}(-1)\oplus L_{C}$ and
$\mathcal{V}_{2}\cong\mathcal{T}\oplus(V/\breve{W})_{C}$ for the induced
splittings. The Euler sections decompose as
$\mathrm{eu}_{\pi_{1}}=(u_{1},u_{1}^{\prime})^{\vee}$ and
$\mathrm{eu}_{\pi_{2}}=(u_{2}^{\prime},u_{2})^{\vee}$, and their vanishing
loci are:

$\displaystyle\mathrm{V}(u_{1})$ $\displaystyle=\set{(y\mapsto
y_{0}\in\ell_{0})\in\mathbf{P}:y=\infty},$
$\displaystyle\mathrm{V}(u_{2}^{\prime})$ $\displaystyle=\set{(y\mapsto
y_{0}\in\ell_{0})\in\mathbf{P}:\mathbf{P}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{U}\in\ell_{0}},$ $\displaystyle\mathrm{V}(u_{1}^{\prime})$
$\displaystyle=\set{(y\mapsto
y_{0}\in\ell_{0})\in\mathbf{P}:y\in\mathbf{P}W},$
$\displaystyle\mathrm{V}(u_{2})$ $\displaystyle=\set{(y\mapsto
y_{0}\in\ell_{0})\in\mathbf{P}:\ell_{0}\subset\mathbf{P}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{W}}.$

Begin by describing the boundary of $T^{\circ}$ in $T$:

######  3.1 Proposition. —

The boundary $T\cap(\mathbf{P}\setminus\mathbf{P}^{\circ})$ is the union of
the effective Cartier divisors

$\displaystyle C^{\prime}$ $\displaystyle\coloneqq\big{\\{}(\infty\mapsto
y_{0}\in\ell_{0})\in\mathbf{P}:y_{0}=\mathbf{P}L_{0}\in
C\;\text{and}\;\ell_{0}^{[1]}=\mathbf{P}L_{0}^{\perp}\big{\\}}\;\;\text{and}$
$\displaystyle D$ $\displaystyle\coloneqq\big{\\{}(y\mapsto
y_{0}\in\mathbf{T}_{C,y_{0}})\in\mathbf{P}:y_{0}\in\mathrm{C}_{\mathrm{Herm}}\;\text{and}\;y\in\langle
y_{0},\infty\rangle\big{\\}}.$

Furthermore, $C^{\prime}$ is a purely inseparable multisection of degree $q$,
and $D$ is a disjoint union of $q^{3}+1$ smooth rational curves.

###### Proof.

Consider the intersection between $T^{\prime}=\mathrm{V}(v)$ with the
irreducible components $Z_{i}\coloneqq\mathrm{V}(u_{i})$ of
$Z\coloneqq\mathbf{P}\setminus\mathbf{P}^{\circ}=\mathrm{V}(u_{1}u_{2})$. The
computation of 2.9 gives the first equalities in

$\displaystyle T^{\prime}\cap Z_{1}$
$\displaystyle=(u_{1}=u_{2}\cdot\beta_{2}^{\vee}\cdot u_{1}^{\prime
q}=u_{2}^{q}\cdot\beta_{2}\cdot u_{1}^{\prime}=0)=Z_{1}\cap
Z_{2},\;\text{and}$ $\displaystyle T^{\prime}\cap Z_{2}$
$\displaystyle=(u_{2}=u_{2}^{\prime}\cdot\beta_{1}^{\vee}\cdot
u_{1}^{q}=u_{2}^{\prime q}\cdot\beta_{1}\cdot u_{1}=0)=(Z_{1}\cap
Z_{2})\cup(u_{2}=u_{2}^{\prime}\cdot\beta_{1}^{\vee}=u_{2}^{\prime
q}\cdot\beta_{1}=0).$

The second equality is then clear for $T^{\prime}\cap Z_{2}$. As for
$T^{\prime}\cap Z_{1}$, it is because $u_{1}^{\prime}$ and $u_{1}$ do not
vanish simultaneously, and $\beta_{2}^{\vee}\colon
L_{C}^{[1]}\to(V/\breve{W})_{C}^{\vee}$ is an isomorphism by 2.1(ii). Next,
2.9 implies that

$-\det(v^{\prime})\rvert_{T^{\prime}\cap Z}=u_{2}^{\prime
q}\cdot\beta_{1}\cdot\beta_{2}^{\vee}\cdot u_{1}^{\prime q}.$

This cuts out the locus $(u_{1}=u_{2}=u_{2}^{\prime q}\cdot\beta_{1}=0)$ on
$Z_{1}\cap Z_{2}$ and vanishes on the second component of $T^{\prime}\cap
Z_{2}$. It remains to identify these with $C^{\prime}$ and $D$, respectively.
The value of

$u_{2}^{\prime
q}\cdot\beta_{1}\colon\mathcal{O}_{C}(-1)\to\mathcal{T}^{[1],\vee}\to\mathcal{O}_{\mathbf{P}}(0,q)$

at a point $(y\mapsto y_{0}\in\ell_{0})$ is determined as follows: a local
generator of $\mathcal{O}_{C}(-1)$ corresponds to a basis vector $v$ of the
subspace $L_{0}\subset\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{W}$
underlying $y_{0}$; $\beta_{1}$ maps this to the linear functional
$\beta_{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{W}}(-,v)$ on $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{W}^{[1]}$; and the quotient map corresponds to restricting this to the
linear space underlying $\ell_{0}$. Thus this vanishes if and only if
$\ell_{0}=\mathbf{P}L_{0}^{\perp,[-1]}$, and so

$(u_{1}=u_{2}=u_{2}^{\prime q}\cdot\beta_{1}=0)=\big{\\{}(\infty\mapsto
y_{0}\in\ell_{0})\in\mathbf{P}:y_{0}=\mathbf{P}L_{0}\in
C\;\text{and}\;\ell_{0}=\mathbf{P}L_{0}^{\perp,[-1]}\big{\\}},$

which is precisely $C^{\prime}$. Similarly,
$u_{2}^{\prime}\cdot\beta_{1}^{\vee}$ vanishes at points $(y\mapsto
y_{0}\in\ell_{0})$ when $\ell_{0}=\mathbf{P}L_{0}^{[1],\perp}$. Therefore

$(u_{2}=u_{2}^{\prime}\cdot\beta_{1}^{\vee}=u_{2}^{\prime
q}\cdot\beta_{1}=0)=\big{\\{}(y\mapsto
y_{0}\in\ell_{0})\in\mathbf{P}:y_{0}=\mathbf{P}L_{0}\in
C\;\text{and}\;\ell_{0}=\mathbf{P}L_{0}^{[1],\perp}=\mathbf{P}L_{0}^{\perp,[-1]}\big{\\}}.$

This is $D$ since $L_{0}^{[1],\perp}=L_{0}^{\perp,[-1]}$ if and only if
$y_{0}$ is a Hermitian point of $C$, see 1.6.

Both $C^{\prime}$ and $D$ are effective Cartier divisors on $T$: The analysis
shows that $C^{\prime}=T\cap Z_{1}$ and $C^{\prime}+D=T\cap Z_{2}$; since $D$
is the difference of effective Cartier divisors, it is Cartier itself. The
moduli description easily shows that $D$ is a union of fibres of
$\mathbf{P}\mathcal{V}_{1}$ over the $q^{3}+1$ Hermitian points of $C$. To see
that $C^{\prime}$ is a multisection, let $\phi_{C}\colon C\to C$ be the map
that sends a point $y_{1}$ to the residual intersection point with its tangent
line $\mathbf{T}_{C,y_{1}}$, as in 1.6. I claim that there exists a morphism

$\phi_{C}^{\prime}\colon C\to C^{\prime}\colon
y_{1}\mapsto(\infty\mapsto\phi_{C}(y_{1})\in\mathbf{T}_{C,y_{1}}).$

If $y_{1}=\mathbf{P}L_{1}$, then $L_{0}\coloneqq\sigma_{\beta}(L_{1}^{[2]})$
underlies $\phi_{C}(y_{1})$. Since
$\sigma_{\beta}=\beta^{-1}\circ\beta^{[1],\vee}$, the diagram

${V^{[1]}}$${V^{\vee}}$${L_{0}^{\vee}}$${V^{[1]}}$${V^{[2],\vee}}$${L_{1}^{[2],\vee}}$$\scriptstyle{\beta^{\vee}}$$\scriptstyle{\sigma_{\beta}^{\vee}}$$\scriptstyle{\sigma_{\beta}^{\vee}}$$\scriptstyle{\beta^{[1]}}$

commutes, and it follows that $\mathbf{T}_{C,y_{1}}^{[1]}$ has underlying
linear space $L_{0}^{\perp}$ and that $\phi_{C}^{\prime}$ exists. Observe that
the projection $C^{\prime}\to C$ and $\phi_{C}^{\prime}$ are both of degree
$q$, the latter because $\mathbf{T}_{C,y_{1}}$ intersects $C$ at $y_{1}$
generically with multiplicity $q$. Since $\phi_{C}$ is of purely inseparable
of degree $q^{2}$, it follows that $\phi_{C}^{\prime}$ is surjective, and so
$C^{\prime}$ is a purely inseparable multisection. ∎

######  3.2 Corollary. —

$T$ is the Zariski closure of $T^{\circ}$ in $\mathbf{P}$.

###### Proof.

If not, then some irreducible component of $T\setminus T^{\circ}$ would be an
irreducible component of $T$. But this is impossible: on the one hand,
$T\setminus T^{\circ}$ is of pure dimension $1$ by 3.1; on the other hand, $T$
is connected and Cohen–Macaulay by 2.10 and is therefore equidimension $2$ by
[Stacks, 00OV]. ∎

######  3.3 Corollary. —

The map $\rho\colon\tilde{S}\to T$ is surjective and finite flat of degree
$q$, $\tilde{S}$ is the vanishing locus of $v_{3}$ in
$\mathbf{P}\mathcal{V}_{T}$, and there is a short exact sequence of bundles on
$T$ given by

$0\to\mathcal{O}_{T}\to\rho_{*}\mathcal{O}_{\tilde{S}}\to\operatorname{Div}^{q-2}(\mathcal{V}_{T})(1,-2)\otimes\pi^{*}\mathcal{O}_{C}(-1)\otimes
L\to 0.$

In particular, $\rho_{*}\mathcal{O}_{\tilde{S}}$ has an increasing filtration
whose graded pieces are

$\operatorname{gr}_{i}(\rho_{*}\mathcal{O}_{\tilde{S}})=\begin{dcases*}\mathcal{O}_{T}&if
$i=0$,\\\ (\pi^{*}\mathcal{O}_{C}(-q+i)\otimes L^{\otimes q-i})(q-i,-i-1)&if
$1\leq i\leq q-1$.\end{dcases*}$

###### Proof.

Since $T$ is the closure of $T^{\circ}$ by 3.2, the map
$\rho\colon\tilde{S}\to T$ is proper and dominant, whence surjective; 2.15 now
implies $\rho$ is flat of degee $q$, and that $\tilde{S}$ is the vanishing
locus of $v_{3}$. Then 2.14 gives a short exact sequence of sheaves on
$\mathbf{P}\mathcal{V}_{T}$:

$0\to\mathcal{O}_{\rho}(-q)\otimes\rho^{*}\mathcal{O}_{T}(0,-1)\xrightarrow{v_{3}}\mathcal{O}_{\mathbf{P}\mathcal{V}_{T}}\to\mathcal{O}_{\tilde{S}}\to
0.$

Pushing this along $\rho$ gives a short exact sequence of
$\mathcal{O}_{T}$-modules

$0\to\mathcal{O}_{T}\to\rho_{*}\mathcal{O}_{\tilde{S}}\to\mathbf{R}^{1}\rho_{*}\mathcal{O}_{\rho}(-q)\otimes\mathcal{O}_{T}(0,-1)\to
0.$

The Euler sequence gives
$\omega_{\rho}\cong\rho^{*}\det(\mathcal{V}_{T}^{\vee})\otimes\mathcal{O}_{\rho}(-2)$
and so by Grothendieck duality

$\displaystyle\mathbf{R}^{1}\rho_{*}\mathcal{O}_{\rho}(-q)\cong\mathbf{R}\rho_{*}\mathbf{R}\mathcal{H}\\!\mathit{om}_{\mathcal{O}_{\mathbf{P}\mathcal{V}_{T}}}(\mathcal{O}_{\rho}(q)\otimes\omega_{\rho},\omega_{\rho})$
$\displaystyle\cong\mathbf{R}\mathcal{H}\\!\mathit{om}_{\mathcal{O}_{T}}(\mathbf{R}\rho_{*}\mathcal{O}_{\rho}(q-2)\otimes\det(\mathcal{V}_{T}^{\vee}),\mathcal{O}_{T})$
$\displaystyle\cong\operatorname{Div}^{q-2}(\mathcal{V}_{T})\otimes\det(\mathcal{V}_{T}).$

Since $\det(\mathcal{V}_{T})\cong(\pi^{*}\mathcal{O}_{C}(-1)\otimes L)(1,-1)$,
this gives the exact sequence in the statement; the filtration comes from
applying divided powers to the short exact sequence for $\mathcal{V}_{T}$
preceding 2.14. ∎

The map $\rho\colon\tilde{S}\to T$ is a quotient map. To explain, let $G$ be
the subgroup scheme of $\mathbf{GL}_{V}$ which preserves both the flag
$L\subset\breve{W}\subset V$ and the $q$-bic form $\beta$, and induces the
identity on $\breve{W}$ and $V/\breve{W}$.

######  3.4 Lemma. —

If $(X,\infty,\mathbf{P}\breve{W})$ is a smooth cone situation, then

$G\cong\begin{dcases*}\mathbf{F}_{q}&if $(X,\infty,\mathbf{P}\breve{W})$ is as
in {\bf\ref{cone-situation-examples}}\ref{cone-situation-examples.smooth},
and\\\ \boldsymbol{\alpha}_{q}&if $(X,\infty,\mathbf{P}\breve{W})$ is as in
{\bf\ref{cone-situation-examples}}\ref{cone-situation-
examples.N2-}.\end{dcases*}$

###### Proof.

A point $g$ of $G$ induces the identity on $\breve{W}$, so
$\delta_{g}\coloneqq g-\operatorname{id}_{V}$ descends to a map
$V/\breve{W}\to V$; since $g$ also induces the identity on $V/\breve{W}$,
$\delta_{g}$ factors as a map $V/\breve{W}\to\breve{W}$. Thus the assignment
$g\mapsto\delta_{g}$ yields a closed immersion $\delta\colon
G\to\operatorname{\mathbf{Hom}}(V/\breve{W},\breve{W})$, the latter viewed as
a vector group. In fact, $\delta$ factors through the algebraic subgroup
$\operatorname{\mathbf{Hom}}(V/\breve{W},L)$: that $G$ preserves $\beta$ and
acts as the identity on $\breve{W}$ together means that

$\beta(\delta_{g}(v)^{[1]},w)=\beta(w^{\prime},\delta_{g}(v))=0$

for every $\mathbf{k}$-algebra $A$, $g\in G(A)$, $v\in
V\otimes_{\mathbf{k}}A$, $w\in\breve{W}\otimes_{\mathbf{k}}A$, and
$w^{\prime}\in(\breve{W}\otimes_{\mathbf{k}}A)^{[1]}$. Splitting
$\breve{W}\cong W\oplus L$ as in 2.9 then shows that $\delta_{g}(v)\in
L\otimes_{\mathbf{k}}A$.

Construct an equation of $G$ in $\operatorname{\mathbf{Hom}}(V/\breve{W},L)$
as follows: Fix a nonzero $w\in L$, and choose $v\in V$ such that its image
$\bar{v}\in V/\breve{W}$ is nonzero, so that $(\bar{v}\mapsto t\cdot w)\mapsto
t$ is an isomorphism
$\operatorname{\mathbf{Hom}}(V/\breve{W},L)\cong\mathbf{G}_{a}$. If
$\delta_{g}$ corresponds to $t\in\mathbf{G}_{a}(A)$ in this way, then

$0=\beta((g\cdot v)^{[1]},g\cdot v)-\beta(v^{[1]},v)\pm\beta(v^{[1]},g\cdot
v)=\beta(w^{[1]},v)t^{q}+\beta(v^{[1]},w)t.$

Since $\infty$ is a smooth point, $L^{[1],\perp}=\breve{W}$ as explained in
2.1, and so $\beta(w^{[1]},v)$ is a nonzero scalar. The scalar
$\beta(v^{[1]},w)$ is nonzero in 2.2(i), and so $G\cong\mathbf{F}_{q}$;
whereas it is zero in 2.2(ii), and so $G\cong\boldsymbol{\alpha}_{q}$. ∎

The linear action of $G$ on $V$ induces an action on the schemes under
consideration. First, it is straightforward that this action is trivial on $C$
and $T$: For $C$, this is because $G$ acts trivially on $\breve{W}$ and fixes
$L$. For $T$, its points are triples $(y\mapsto y_{0}\in\ell_{0})$ where
$y_{0}\in C$, $y\in\mathbf{P}\breve{W}$, and
$\ell_{0}\subset\mathbf{P}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{V}$
intersects $C$ at $y_{0}$; since $G$ moves neither $y_{0}$ nor $y$, and since
$G$ maps $V/\breve{W}$ to $L$ as in the proof of 3.4, it does not move
$\ell_{0}$. Next, comparing with the description in 2.14, this implies that
$G$ fixes the subbundle $\mathbf{P}(\mathcal{T}_{\pi_{1}}(-1,0))$ in
$\mathbf{P}\mathcal{V}_{T}$. Finally, the action of $G$ on $\tilde{S}$ is as
follows:

######  3.5 Lemma. —

The morphism $\rho\colon\tilde{S}\to T$ is the quotient map for $G$.

###### Proof.

The unipotent algebraic group $\operatorname{\mathbf{Hom}}(V/\breve{W},L)$
acts freely on the open subscheme of $\mathbf{G}$ parameterizing lines $\ell$
satisfying $\infty\notin\ell$ and $\ell\not\subset\mathbf{P}\breve{W}$. The
proof of 3.4 shows that $G$ is a closed subgroup scheme of this unipotent
group, and so $G$ acts freely on the open subscheme $S^{\circ}$ of $S$. The
result follows upon identifying $S^{\circ}$ with the open subscheme
$\tilde{S}$ consisting of points $((y\in\ell)\mapsto(y_{0}\in\ell_{0}))$ where
$y\neq\infty$ and $\ell\not\subset\mathbf{P}\breve{W}$: Indeed the canonical
morphism $\tilde{S}/G\to T$ is an isomorphism over $T^{\circ}$. Since
$\tilde{S}\to T$ is surjective and finite of degree $q$ by 3.3, and lengths of
fibres of finite morphisms are upper semicontinuous by Nakayama,
$\tilde{S}/G\to T$ has degree $1$, and so is an isomorphism. ∎

Putting 2.7, 3.1, and 3.3 together identifies the points of the complement
$S\setminus S^{\circ}$ as follows:

######  3.6 Lemma. —

The complement of $S^{\circ}$ in $\tilde{S}$ the union of effective Cartier
divisors

$\displaystyle\tilde{C}_{\infty}$
$\displaystyle\coloneqq\big{\\{}((\infty\in\ell)\mapsto(y_{0}\in\ell_{0}))\in\tilde{S}:y_{0}=\mathbf{P}L_{0}\in
C,\;\ell_{0}^{[1]}=\mathbf{P}L_{0}^{\perp},\;\ell=\langle
y_{0},\infty\rangle\big{\\}}\;\text{and}$ $\displaystyle E$
$\displaystyle\coloneqq\big{\\{}((y\in\ell)\mapsto(y_{0}\in\mathbf{T}_{C,y_{0}}))\in\tilde{S}:y_{0}\in
C_{\mathrm{Herm}}\,\;\text{and}\;\,\ell=\langle y_{0},\infty\rangle\big{\\}}.$

###### Proof.

It remains to prove that $\tilde{C}_{\infty}$ and $E$ are Cartier in
$\tilde{S}$. Let $C_{\infty}$ be the closed subscheme of $S$ parameterizing
lines through $\infty$; it follows from 2.4(ii) that it is isomorphic to the
smooth $q$-bic curve $C$, and that, by 1.8, $S$ is smooth along $C_{\infty}$.
Therefore the description of its points shows that $\tilde{C}_{\infty}$ is the
strict transform of the effective Cartier divisor $C_{\infty}$, and so it,
too, is effective Cartier. Since $\tilde{C}_{\infty}+E$ is the vanishing locus
of the section $u_{3}$ from 2.14, it follows that $E$ is also Cartier. ∎

Observing that the Hermitian points of $C_{\infty}$ parameterize lines $\ell$
in $X$ which project to Hermitian points of $C$ and comparing with 3.6
essentially gives:

######  3.7 Corollary. —

The morphism $\tilde{S}\to S$ is a blowup along the Hermitian points of
$C_{\infty}$.

###### Proof.

The Hermitian points of $C_{\infty}$ pullback to the effective Cartier divisor
$E$ in $\tilde{S}$, so the blowup $S^{\prime}\to S$ along these points admits
a canonical morphism $S^{\prime}\to\tilde{S}$. Since $S$ is smooth along
$C_{\infty}$, Zariski’s Main Theorem as in [Har77, Corollary III.11.4] applies
to show that $\tilde{S}\to S^{\prime}$ is an isomorphism. ∎

In particular, this implies that $\tilde{S}$ and $T$ are smooth along their
boundary:

######  3.8 Corollary. —

$\tilde{S}$ is smooth along $\tilde{C}_{\infty}\cup E$, and $T$ is smooth
along $C^{\prime}\cup D$.

###### Proof.

Smoothness of $\tilde{S}$ along $\tilde{C}_{\infty}\cup E$ follows from that
of $S$ along $C_{\infty}$; that of $T$ along $C^{\prime}\cup D$ is because
$\rho\colon\tilde{S}\to T$ is flat by 3.3, and smoothness descends along flat
morphisms, see [Stacks, 05AW]. ∎

######  3.9 Corollary. —

The rational map $S\dashrightarrow T$ is defined away from the Hermitian
points of $C_{\infty}$, and the rational map $\varphi\colon S\dashrightarrow
C$ extends to a morphism.

###### Proof.

The rational maps from $S$ to both $T$ and $C$ are resolved up on $\tilde{S}$,
so the statement about $S\dashrightarrow T$ follows directly from 3.7, and
that about $\varphi\colon S\dashrightarrow C$ follows from 3.6 which implies
that each component of the exceptional divisor $E$ is mapped to a single point
along $\tilde{S}\to C$. ∎

Since the boundary divisor $Z$ in $\mathbf{P}$ is relatively ample and since
$T\cap Z\to C$ has connected fibres by 3.1, the fibres of $T\to C$ are
connected. Using 3.5 and 3.6 then implies the same about the fibres of
$\tilde{S}\to C$, and then 3.7 implies the same for $S\to C$. This almost
implies that, for instance, $\varphi_{*}\mathcal{O}_{S}\cong\mathcal{O}_{C}$,
but there is the matter of reduced fibres and the possibility of a factoring
through a purely inseparable cover of $C$. At any rate, the following
clarifies the structure of $\mathbf{R}\varphi_{*}\mathcal{O}_{S}$:

######  3.10 Lemma. —

The natural maps give isomorphisms

$\varphi_{*}\mathcal{O}_{S}\cong(\pi\circ\rho)_{*}\mathcal{O}_{\tilde{S}}\cong\pi_{*}\mathcal{O}_{T}\cong\mathcal{O}_{C},$

and $\mathbf{R}^{1}\varphi_{*}\mathcal{O}_{S}$ is locally free and carries a
filtration with graded pieces

$\mathrm{gr}_{i}(\mathbf{R}^{1}\varphi_{*}\mathcal{O}_{S})\cong\begin{dcases*}\mathbf{R}^{1}\pi_{*}\mathcal{O}_{T}&if
$i=0$, and\\\ \mathcal{O}_{C}(-q+i)\otimes L^{\otimes
q-i}\otimes\mathbf{R}^{1}\pi_{*}\mathcal{O}_{T}(q-i,-i-1)&if $1\leq i\leq
q-1$.\end{dcases*}$

###### Proof.

By 2.10, the structure sheaf of $T$ admits a resolution on $\mathbf{P}$ of the
form

$0\to\mathcal{E}_{2}(-q-1,-q-1)\to\mathcal{E}_{1}(-q-1,-q-1)\to\mathcal{O}_{\mathbf{P}}\to\mathcal{O}_{T}\to
0.$

The twists of $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ are sums of negative
line bundles on a $\mathbf{P}^{1}\times\mathbf{P}^{2}$-bundle over $C$, and so
cohomology is in degree $3$. Therefore the spectral sequence computing
cohomology of $\mathcal{O}_{T}$ shows that the natural map
$\mathcal{O}_{C}\to\pi_{*}\mathcal{O}_{T}$ is an isomorphism.

By 3.3, $\rho_{*}\mathcal{O}_{\tilde{S}}$ has a filtration with graded pieces
$\mathcal{O}_{T}$ and
$\mathcal{O}_{T}(q-i,-i-1)\otimes\pi^{*}\mathcal{O}_{C}(-q+i)$ where $1\leq
i\leq q-1$. The resolution above implies that the latter terms have vanishing
pushforward: $\mathcal{E}_{1}(-i-1,-q-i-2)$ consists of line bundles which are
negative on the $\mathbf{P}^{2}$-side of $\mathbf{P}\to C$, so cohomology is
supported in degrees $2$ and $3$; whereas $\mathcal{E}_{2}(-i-1,-q-i-2)$ is
negative on both factors and so cohomology is supported in degree $3$.
Therefore $\mathcal{O}_{C}\to(\pi\circ\rho)_{*}\mathcal{O}_{\tilde{S}}$ is an
isomorphism. Finally, since $\tilde{S}\to S$ is a blowup at smooth point by
3.7, it follows that
$\mathbf{R}\varphi_{*}\mathcal{O}_{S}\cong\mathbf{R}(\pi\circ\rho)_{*}\mathcal{O}_{\tilde{S}}$
and so $\mathcal{O}_{C}\to\varphi_{*}\mathcal{O}_{S}$ is an isomorphism,
$\mathbf{R}^{1}\varphi_{*}\mathcal{O}_{S}$ is locally free by cohomology and
base change, and carries the filtration by 3.3. ∎

##  4\. $q$-bic threefolds of type $\mathbf{1}^{\oplus 3}\oplus\mathbf{N}_{2}$

This and the following two sections are concerned with the geometry of mildly
singular $q$-bic threefolds, namely, those of type $\mathbf{1}^{\oplus
3}\oplus\mathbf{N}_{2}$. The purpose of this section is to construct the
normalization $\nu\colon S^{\nu}\to S$ of the Fano surface, see 4.7, and to
relate the cohomology of $\mathcal{O}_{S}$ with that of an
$\mathcal{O}_{C}$-module $\mathcal{F}$ related to the quotient
$\nu_{*}\mathcal{O}_{S^{\nu}}/\mathcal{O}_{S}$, see 4.13 and 4.14.

### 4.1.

— Throughout §§4–6, let $(V,\beta)$ be a $q$-bic form of type
$\mathbf{1}^{\oplus 3}\oplus\mathbf{N}_{2}$, and let $X$ be the associated
$q$-bic threefold. The two kernels $L_{+}\coloneqq V^{\perp,[-1]}$ and
$L_{-}\coloneqq V^{[1],\perp}$ of $\beta$ underlie the singular point $x_{+}$
and the special smooth point $x_{-}$ of $X$. Setting $U\coloneqq L_{-}\oplus
L_{+}$, the classification of $q$-bic forms provides a canonical orthogonal
decomposition $V=W\oplus U$ where $\beta_{W}$ is of type $\mathbf{1}^{\oplus
3}$. The plane $\mathbf{P}W$ intersects $X$ at the smooth $q$-bic curve $C$
defined by $\beta_{W}$; this curve is canonically identified as the base of
the cones $X_{-}$ and $X_{+}$ in the cone situations

$(X,x_{-},\mathbf{P}L_{-}^{[1],\perp})\quad\text{and}\quad(X,x_{+},\mathbf{P}L_{+}^{\perp,[-1]})$

as in 2.2(ii) and 2.2(iii).

The Fano scheme $S$ of lines in $X$ is of expected dimension $2$: this
follows, for example, from 2.3(i) applied to either cone situation above.
Alternatively, by 2.4(i) and 2.4(ii), the subschemes $C_{\pm}\subset S$
parameterizing lines through the special points $x_{\pm}\in X$ are supported
on a scheme isomorphic to $C$, so the singular locus of $S$ has dimension $1$,
and the discussion of 1.8 shows $S$ is a surface.

###  4.2. Automorphisms

— Consider the automorphism group scheme
$\operatorname{\mathbf{Aut}}(V,\beta)$ of the $q$-bic form, as introduced in
[Che23a, 5.1]. Specializing the computation of [Che22, 1.3.7] with $a=1$ and
$b=3$ gives the following explicit description as a closed sub-group scheme of
$\mathbf{GL}_{5}$:

$\operatorname{\mathbf{Aut}}(V,\beta)\cong\Set{\left(\begin{array}[]{c@{}}{cc}A&0&\mathbf{x}\\\
\hline\cr\mathbf{y}^{\vee}&\lambda^{-1}&\epsilon\\\
0&0&\lambda^{q}\end{array}\right)\in\mathbf{GL}_{5}|\begin{array}[]{ccc}\lambda\in\mathbf{G}_{m},&\mathbf{x}\in\boldsymbol{\alpha}^{3}_{q\phantom{{}^{2}}},&A\in\mathrm{U}_{3}(q),\\\
\epsilon\in\boldsymbol{\alpha}_{q},&\mathbf{y}\in\boldsymbol{\alpha}_{q^{2}}^{3},&\lambda^{q}A^{\vee,[1]}\mathbf{x}=\mathbf{y}^{[1]}\end{array}}{.}$

A particularly useful subgroup is that which preserves the orthogonal
decomposition $V=W\oplus U$:

$\mathrm{G}\coloneqq\operatorname{\mathbf{Aut}}(W,\beta_{W})\times\operatorname{\mathbf{Aut}}(L_{-}\subset
U,\beta_{U})\cong\mathrm{U}_{3}(q)\times\Set{\begin{pmatrix}\lambda^{-1}&\epsilon\\\
0&\lambda^{q}\end{pmatrix}:\lambda\in\mathbf{G}_{m},\epsilon\in\boldsymbol{\alpha}_{q}}$

Observe that the torus $\mathbf{G}_{m}$ in
$\operatorname{\mathbf{Aut}}(V,\beta)$ acts, in particular, on $X$ and the
Fano surface $S$; it is straightforward to check that its fixed schemes are
$X^{\mathbf{G}_{m}}=\\{x_{-},x_{+}\\}\cup C$ and $S^{\mathbf{G}_{m}}=C_{-}\cup
C_{+}$, respectively.

###  4.3. Cone situations

— Both cone situations in 4.1 associated with the special points $x_{\mp}\in
X$ satisfy 2.1(i) and 2.1(iii), so 2.5 gives rational maps

$\varphi_{-}\colon S\longrightarrow C\quad\text{and}\quad\varphi_{+}\colon
S\dashrightarrow C,$

where the orthogonal decomposition $V=W\oplus U$ identifies the target as the
curve $C=X\cap\mathbf{P}W$. Since $\varphi_{-}$ arises from a smooth cone
situation, it extends to a morphism by 3.9, whereas $\varphi_{+}$ is defined
only away from $C_{+}$. Writing $\operatorname{proj}_{\mathbf{P}U}\colon
X\dashrightarrow\mathbf{P}W$ for the rational map induced by linear projection
centred at $\mathbf{P}U$, the specific geometry of $X$ offers an alternative
description of the maps $\varphi_{\mp}$:

######  4.4 Lemma. —

Let $\ell\subset X$ be a line not passing through either $x_{\mp}$. Then
$\ell_{0}\coloneqq\operatorname{proj}_{\mathbf{P}U}(\ell)$ is a line in
$\mathbf{P}W$ tangent to $C$ at the point
$\operatorname{proj}_{\mathbf{P}U}(\ell\cap X_{+})$ with residual intersection
point $\operatorname{proj}_{\mathbf{P}U}(\ell\cap X_{-})$, so

$\displaystyle\varphi_{+}([\ell])$ $\displaystyle=\text{point of tangency
between $\ell_{0}$ and $C$},$ $\displaystyle\varphi_{-}([\ell])$
$\displaystyle=\text{residual point of intersection between $\ell_{0}$ and
$C$}.$

###### Proof.

Projection from $\mathbf{P}U$ contracts only lines through $x_{\mp}$, so
$\ell_{0}$ is a line in $\mathbf{P}W$. Since $\mathbf{P}U$ intersects $X$ only
at the vertices $x_{\mp}$ of the two cones over $C$, and since $x_{+}$ is a
singular point of multiplicity $q$,

$C\cap\ell_{0}=\operatorname{proj}_{\mathbf{P}U}\big{(}\ell\cap\operatorname{proj}_{\mathbf{P}U}^{-1}(C)\big{)}=\operatorname{proj}_{\mathbf{P}U}\big{(}\ell\cap(X_{-}\cup
qX_{+})\big{)}.$

Thus $\ell_{0}$ and $C$ are tangent at
$\operatorname{proj}_{\mathbf{P}U}(\ell\cap X_{+})$ and have residual point of
intersection at $\operatorname{proj}_{\mathbf{P}U}(\ell\cap X_{-})$. On the
other hand, the definition of $\varphi_{\mp}$ from 2.5 gives

$\varphi_{\mp}([\ell])=\operatorname{proj}_{x_{\mp}}(\ell\cap
X_{\mp})=\operatorname{proj}_{\mathbf{P}U}(\ell\cap X_{\mp})$

with the second equality because $x_{\pm}\notin X_{\mp}$. ∎

With more notation, the proof works for families of lines in
$X\setminus\\{x_{-},x_{+}\\}$. This leads to another proof of 3.9, that
$\varphi_{-}$ extends over $C_{-}$: 4.4 means that $\varphi_{-}([\ell])$ is
the residual intersection point of $C$ with its tangent line at
$\operatorname{proj}_{\mathbf{P}U}(\ell\cap X_{+})$, and this makes sense even
when $x_{-}\in\ell$. More interestingly, this gives an alternative moduli
interpretation of $S\setminus C_{+}$ as a scheme over $C$ via $\varphi_{+}$.
To describe it, let $\mathcal{E}_{C}$ be the _embedded tangent bundle_ $C$ in
$\mathbf{P}W$, which is defined via pullback of the right square in the
following commutative diagram of short exact sequences:

${0}$${\mathcal{O}_{C}(-1)}$${\mathcal{E}_{C}}$${\mathcal{T}_{C}(-1)}$${0}$${0}$${\mathcal{O}_{C}(-1)}$${W_{C}(-1)}$${\mathcal{T}}$${0}$$=$$\subset$$\subset$

where, as in 2.6,
$\mathcal{T}\coloneqq\mathcal{T}_{\mathbf{P}W}(-1)\rvert_{C}$. The fibre of
$\mathcal{E}_{C}$ over a point $y_{0}=\mathbf{P}L_{0}$ is the subspace
$L_{0}^{[1],\perp}$ in $W$ underlying the embedded tangent space
$\mathbf{T}_{C,y_{0}}$, see 1.5. The subbundle
$\mathcal{W}\coloneqq\mathcal{E}_{C}\oplus U_{C}$ of $V_{C}$ defines the
family $\mathbf{P}\mathcal{W}\to C$ of hyperplanes in $\mathbf{P}V$ spanned by
the tangent lines to $C$ and $U$. Let $X_{\mathbf{P}\mathcal{W}}\coloneqq
X\times_{\mathbf{P}V}\mathbf{P}\mathcal{W}$ be the corresponding family of
hyperplane sections of $X$. A reformulation of 4.4 in these terms is then:

######  4.5 Corollary. —

Let $\mathbf{L}^{\circ}$ be the restriction of the Fano correspondence
$S\leftarrow\mathbf{L}\rightarrow X$ to $S\setminus C_{+}$. Then the morphism
$\mathbf{L}^{\circ}\to X$ factors through $X_{\mathbf{P}\mathcal{W}}$ and fits
into a commutative diagram

${\mathbf{L}^{\circ}}$${X_{\mathbf{P}\mathcal{W}}}$${X}$${S\setminus
C_{+}}$${C\makebox[0.0pt][l]{\,.}}$$\scriptstyle{\varphi_{+}}$

###### Proof.

Away from $C_{-}\subset S$, this follows from 4.4. It remains to observe that
the lines through $x_{-}$ correspond to the subbundle
$\mathcal{O}_{C}(-1)\oplus L_{-,C}$ of $\mathcal{W}$. ∎

A line $\ell\subset X\setminus x_{+}$ such that $\varphi_{+}([\ell])=y_{0}$
intersects $X_{+}$ along the line $\langle x_{+},y_{0}\rangle$. These lines
are globally parameterized by the subbundle
$\mathcal{W}^{\prime}\coloneqq\mathcal{O}_{C}(-1)\oplus L_{+,C}$; together
with 4.4 and 4.5, this implies that
$\mathbf{L}^{\circ}\times_{X_{\mathbf{P}\mathcal{W}}}\mathbf{P}\mathcal{W}^{\prime}$
projects isomorphically to $S\setminus C_{+}$. This suggests that
$\varphi_{+}$ may be resolved by considering all lines in
$X_{\mathbf{P}\mathcal{W}}$ that pass through
$\mathbf{P}\mathcal{W}^{\prime}$. To proceed, consider the morphism
$X_{\mathbf{P}\mathcal{W}}\to C$:

######  4.6 Lemma. —

$X_{\mathbf{P}\mathcal{W}}\to C$ is a family of corank $2$ $q$-bic surfaces
with singular locus $\mathbf{P}\mathcal{W}^{\prime}$.

###### Proof.

This family of $q$-bic surfaces over $C$ is determined by the $q$-bic form
$\beta_{\mathcal{W}}\colon\mathcal{W}^{[1]}\otimes\mathcal{W}\to\mathcal{O}_{C}$
obtained by restricting $\beta$ to $\mathcal{W}$. The orthogonal splitting
$V=W\oplus U$ restricts to a decomposition

$(\mathcal{W},\beta_{\mathcal{W}})=(\mathcal{E}_{C},\beta_{W,\mathrm{tan}})\oplus(U_{C},\beta_{U})$

where $\beta_{W,\mathrm{tan}}$ is $q$-bic form induced by $\beta_{W}$ on the
embedded tangent spaces to $C$. Observe that $\beta_{U}$ is of constant type
$\mathbf{N}_{2}$, and $\beta_{W,\mathrm{tan}}$ is of type
$\mathbf{0}\oplus\mathbf{1}$ or $\mathbf{N}_{2}$ depending on whether or not
the point of tangency is a Hermitian point of $C$; in particular,
$\beta_{\mathcal{W}}$ is everywhere of corank $2$. By [Che23b, 2.6], the
nonsmooth locus of $X_{\mathbf{P}\mathcal{W}}\to C$ corresponds to the
subbundle

$\mathcal{W}^{\perp}=\mathcal{E}_{C}^{\perp_{\beta_{W,\mathrm{tan}}}}\oplus
U_{C}^{\perp_{\beta_{U}}}=\mathcal{O}_{C}(-q)\oplus
L_{+,C}^{[1]}\subset\mathcal{W}^{[1]}$

where the two orthogonals can be computed geometrically by noting that
$X\cap\mathbf{P}U$ is singular at $x_{+}$, and the intersection between $C$
and its tangent line at $y_{0}$ is singular at $y_{0}$. This bundle descends
through Frobenius to $\mathcal{W}^{\prime}$ as in the statement, and so
$X_{\mathbf{P}\mathcal{W}}$ itself is singular along thereon. ∎

Construct the family of lines in $X_{\mathbf{P}\mathcal{W}}$ over $C$ through
$\mathbf{P}\mathcal{W}^{\prime}$ as follows: Set
$\mathcal{W}^{\prime\prime}\coloneqq\mathcal{W}/\mathcal{W}^{\prime}$, let
$S^{\nu}\coloneqq\mathbf{P}\mathcal{W}^{\prime\prime}$ be the associated
$\mathbf{P}^{1}$-bundle, and write $\tilde{\varphi}_{+}\colon S^{\nu}\to C$
for the structure map. Linear projection over $C$ with centre
$\mathbf{P}\mathcal{W}^{\prime}$ produces a rational map
$X_{\mathbf{P}\mathcal{W}}\dashrightarrow S^{\nu}$ which is resolved on the
blowup $\tilde{X}_{\mathbf{P}\mathcal{W}}$ along
$\mathbf{P}\mathcal{W}^{\prime}$. Since $X_{\mathbf{P}\mathcal{W}}$ is a
family of $q$-bic surfaces over $C$, its singular locus has multiplicity $q$,
the fibres of $\tilde{X}_{\mathbf{P}\mathcal{W}}\to S^{\nu}$ are curves of
degree $1$ in $\mathbf{P}V$. In other words, this is a family of lines in $X$,
and so defines a morphism $\nu\colon S^{\nu}\to S$.

######  4.7 Proposition. —

The morphism $\nu\colon S^{\nu}\to S$ is the normalization, fits into a
commutative diagram

${S^{\nu}}$${S}$${C}$$\scriptstyle{\nu}$$\scriptstyle{\tilde{\varphi}_{+}}$$\scriptstyle{\varphi_{+}}$

and satisfies
$\nu^{*}\mathcal{O}_{S}(1)=\tilde{\varphi}_{+}^{*}\mathcal{O}_{C}(1)\otimes\mathcal{O}_{\tilde{\varphi}_{+}}(q+1)\otimes
L_{+}^{\vee}$.

###### Proof.

Note $\mathcal{W}^{\prime\prime}\cong\mathcal{T}_{C}(-1)\oplus L_{-,C}$ and
the fibres of $\tilde{X}_{\mathbf{P}\mathcal{W}}\to S^{\nu}$ over
$C_{+}^{\nu}\coloneqq\mathbf{P}(\mathcal{T}_{C}(-1))$ are those lines through
$x_{+}\in X$. The statement of and the comments following 4.5 imply that
$S\setminus C_{+}$ and $S^{\nu}\setminus C_{+}^{\nu}$ represent the same
moduli problem over $C$, and this implies the first two statements.

To compute the pullback of the Plücker line bundle, describe the
$\mathbf{P}^{1}$-bundle $\tilde{X}_{\mathbf{P}\mathcal{W}}\to S^{\nu}$ more
precisely: The blowup of $\mathbf{P}\mathcal{W}$ along
$\mathbf{P}\mathcal{W}^{\prime}$ is canonically the $\mathbf{P}^{2}$-bundle
over $S^{\nu}$ associated with $\tilde{\mathcal{W}}$ formed in the pullback
diagram

${0}$${\tilde{\varphi}_{+}^{*}\mathcal{W}^{\prime}}$${\tilde{\mathcal{W}}}$${\mathcal{O}_{\tilde{\varphi}_{+}}(-1)}$${0}$${0}$${\tilde{\varphi}_{+}^{*}\mathcal{W}^{\prime}}$${\tilde{\varphi}_{+}^{*}\mathcal{W}}$${\tilde{\varphi}_{+}^{*}\mathcal{W}^{\prime\prime}}$${0}$$\scriptstyle{\mathrm{eu}_{\tilde{\varphi}_{+}}}$

and so the exceptional divisor of
$\mathbf{P}\tilde{\mathcal{W}}\to\mathbf{P}\mathcal{W}$ is the subbundle
$\mathbf{P}(\tilde{\varphi}_{+}^{*}\mathcal{W}^{\prime})\subset\mathbf{P}\tilde{\mathcal{W}}$.
The inverse image $X_{\mathbf{P}\tilde{\mathcal{W}}}\coloneqq
X_{\mathbf{P}\mathcal{W}}\times_{\mathbf{P}\mathcal{W}}\mathbf{P}\tilde{\mathcal{W}}$
of $X_{\mathbf{P}\mathcal{W}}$ along this blowup is the bundle of $q$-bic
curves over $S^{\nu}$ defined by the $q$-bic form
$\beta_{\tilde{\mathcal{W}}}\colon\tilde{\mathcal{W}}^{[1]}\otimes\tilde{\mathcal{W}}\to\mathcal{O}_{S^{\nu}}$
obtained by restricting $\tilde{\varphi}^{*}_{+}\beta_{\mathcal{W}}$. Since
$\mathcal{W}^{\perp}=\mathcal{W}^{\prime[1]}$ by 4.6, the diagram above
implies that
$\tilde{\varphi}_{+}^{*}\mathcal{W}^{\prime[1]}=\tilde{\mathcal{W}}^{\perp}$.
Thus by [Che23a, 1.5], there is an exact sequence

$0\to\mathcal{K}\to\tilde{\mathcal{W}}\xrightarrow{\beta_{\tilde{\mathcal{W}}}}\tilde{\mathcal{W}}^{[1],\vee}\to\tilde{\varphi}_{+}^{*}\mathcal{W}^{\prime[1],\vee}\to
0$

where $\mathcal{K}$ is the rank $2$ subbundle defining the
$\mathbf{P}^{1}$-bundle $\tilde{X}_{\mathbf{P}\mathcal{W}}\to S^{\nu}$. So
since $\nu^{*}\mathcal{S}=\mathcal{K}$,

$\displaystyle\nu^{*}\mathcal{O}_{S}(1)\cong\nu^{*}\det(\mathcal{S})^{\vee}\cong\det(\mathcal{K})^{\vee}$
$\displaystyle\cong\det(\tilde{\mathcal{W}})^{\vee,\otimes
q+1}\otimes\tilde{\varphi}^{*}_{+}\det(\mathcal{W}^{\prime})^{\otimes q}$
$\displaystyle\cong\tilde{\varphi}^{*}_{+}\det(\mathcal{W}^{\prime})^{\vee}\otimes\mathcal{O}_{\tilde{\varphi}_{+}}(q+1)$
$\displaystyle\cong\tilde{\varphi}^{*}_{+}\mathcal{O}_{C}(1)\otimes\mathcal{O}_{\tilde{\varphi}_{+}}(q+1)\otimes
L_{+}^{\vee}$

since $\mathcal{W}^{\prime}=\mathcal{O}_{C}(-1)\oplus L_{+,C}$. ∎

The following statement summarizes 4.4 and 4.7 and gives a direct geometric
relationship between the maps $\tilde{\varphi}_{+}$ and $\varphi_{-}$. For
this, let $\phi_{C}\colon C\to C$ be the endomorphism that sends a point $x$
to the residual intersection point between $C$ and its tangent line at $x$, as
described in 1.6.

######  4.8 Corollary. —

There is a commutative diagram of morphisms

${S^{\nu}}$${S}$${C}$${C\makebox[0.0pt][l]{\,.}}$$\scriptstyle{\nu}$$\scriptstyle{\tilde{\varphi}_{+}}$$\scriptstyle{\varphi_{-}}$$\scriptstyle{\phi_{C}}$

###  4.9. Conductors

— Consider the conductor ideals associated with the normalization $\nu\colon
S^{\nu}\to S$:

$\operatorname{cond}_{\nu,S}\coloneqq\mathcal{A}\\!\mathit{nn}_{\mathcal{O}_{S}}(\nu_{*}\mathcal{O}_{S^{\nu}}/\mathcal{O}_{S})\subset\mathcal{O}_{S}\quad\text{and}\quad\operatorname{cond}_{\nu,S^{\nu}}\coloneqq\nu^{-1}\operatorname{cond}_{\nu,S}\cdot\mathcal{O}_{S^{\nu}}\subset\mathcal{O}_{S^{\nu}}.$

The corresponding conductor subschemes $D\subset S$ and $D^{\nu}\subset
S^{\nu}$ are thickenings of the curves $C_{+}$ and $C_{+}^{\nu}$,
respectively, and fit into a commutative diagram

${D^{\nu}}$${D}$${C}$${C}$$\scriptstyle{\nu}$$\scriptstyle{\tilde{\varphi}_{+}}$$\scriptstyle{\varphi_{-}}$$\scriptstyle{\phi_{C}}$

where the vertical maps are finite by properness of $S$ and $S^{\nu}$ over
$C$. Algebraically, the conductor ideal of $S$ is characterized as the largest
ideal of $\mathcal{O}_{S}$ which is also an ideal of
$\nu_{*}\mathcal{O}_{S^{\nu}}$, so there is a commutative diagram of exact
sequences of sheaves on $S$:

${\operatorname{cond}_{\nu,S}}$${\nu_{*}\operatorname{cond}_{\nu,S^{\nu}}}$${0}$${\mathcal{O}_{S}}$${\nu_{*}\mathcal{O}_{S^{\nu}}}$${\nu_{*}\mathcal{O}_{S^{\nu}}/\mathcal{O}_{S}}$${0}$${0}$${\mathcal{O}_{D}}$${\nu_{*}\mathcal{O}_{D^{\nu}}}$${\nu_{*}\mathcal{O}_{D^{\nu}}/\mathcal{O}_{D}}$${0\makebox[0.0pt][l]{\,.}}$$\scriptstyle{\cong}$$\scriptstyle{\nu^{\\#}}$$\scriptstyle{\cong}$$\scriptstyle{\nu^{\\#}}$

Duality theory for $\nu\colon S^{\nu}\to S$ identifies the conductor ideal
$\operatorname{cond}_{\nu,S^{\nu}}$ with the relative dualizing sheaf
$\omega_{S^{\nu}/S}\cong\omega_{S^{\nu}}\otimes\nu^{*}\omega_{S}^{\vee}$,
compare [Stacks, 0FKW] and [Rei94, Proposition 2.3].

######  4.10 Proposition. —

The conductor ideal of $S^{\nu}$ is isomorphic to

$\operatorname{cond}_{\nu,S^{\nu}}\cong\mathcal{O}_{\tilde{\varphi}_{+}}(-\delta-1)\otimes(L_{+}^{\otimes
2q-1}\otimes L_{-})\quad\text{where}\;\delta\coloneqq 2q^{2}-q-2.$

In particular, the conductor subscheme $D^{\nu}$ is the $\delta$-order
neighbourhood of $C_{+}^{\nu}$.

###### Proof.

Since $S^{\nu}$ is the $\mathbf{P}^{1}$-bundle on
$\mathcal{W}^{\prime\prime}\cong\mathcal{T}_{C}(-1)\oplus L_{-,C}$, as in 4.7,
the relative Euler sequence gives

$\omega_{S^{\nu}}\cong\omega_{S^{\nu}/C}\otimes\tilde{\varphi}_{+}^{*}\omega_{C}\cong\mathcal{O}_{\tilde{\varphi}_{+}}(-2)\otimes\tilde{\varphi}^{*}_{+}(\omega_{C}^{\otimes
2}\otimes\mathcal{O}_{C}(1))\otimes L_{-}^{\vee}.$

By 1.8, $\omega_{S}\cong\mathcal{O}_{S}(2q-3)\otimes(L_{+}\otimes
L_{-})^{\vee,\otimes 2}$, so 4.7 gives

$\displaystyle\nu^{*}\omega_{S}$
$\displaystyle\cong\nu^{*}\mathcal{O}_{S}(2q-3)\otimes(L_{+}\otimes
L_{-})^{\vee,\otimes 2}$
$\displaystyle\cong\tilde{\varphi}_{+}^{*}\mathcal{O}_{C}(2q-3)\otimes\mathcal{O}_{\tilde{\varphi}_{+}}\big{(}(2q-3)(q+1)\big{)}\otimes
L_{+}^{\vee,\otimes 2q-1}\otimes L_{-}^{\vee,\otimes 2}.$

Since $C$ is a plane curve of degree $q+1$, $\omega_{C}^{\otimes
2}\otimes\mathcal{O}_{C}(1)\cong\mathcal{O}_{C}(2q-3)$, and so

$\omega_{S^{\nu}/S}\cong\omega_{S^{\nu}}\otimes\nu^{*}\omega_{S}^{\vee}\cong\mathcal{O}_{\tilde{\varphi}_{+}}(-\delta-1)\otimes(L_{+}^{\otimes
2q-1}\otimes L_{-}).\qed$

###  4.11. The sheaf $\mathcal{F}$

— Let $\mathcal{D}$ and $\mathcal{D}^{\nu}$ be the coherent
$\mathcal{O}_{C}$-algebras by pushing structure sheaves along the finite
morphisms $\varphi_{-}\colon D\to C$ and
$\phi_{C}\circ\tilde{\varphi}_{+}\colon D^{\nu}\to C$, respectively.
Restricting $\nu$ to the conductors induces an injective map
$\nu^{\\#}\colon\mathcal{D}\to\mathcal{D}^{\nu}$ of
$\mathcal{O}_{C}$-algebras. Its cokernel

$\mathcal{F}\coloneqq\mathcal{D}/\mathcal{D}^{\nu}\cong\varphi_{-,*}(\nu_{*}\mathcal{O}_{D^{\nu}}/\mathcal{O}_{D})\cong\varphi_{-,*}(\nu_{*}\mathcal{O}_{S^{\nu}}/\mathcal{O}_{S}),$

identifications arising from the diagram of 4.9, is an
$\mathcal{O}_{C}$-module related to $S$ as follows:

######  4.12 Lemma. —

There is an exact sequence of finite locally free $\mathcal{O}_{C}$-modules

$0\to\mathcal{O}_{C}\to\phi_{C,*}\mathcal{O}_{C}\to\mathcal{F}\to\mathbf{R}^{1}\varphi_{-,*}\mathcal{O}_{S}\to
0$

in which the map $\mathcal{F}\to\mathbf{R}^{1}\varphi_{-,*}\mathcal{O}_{S}$
splits.

###### Proof.

Push the middle row in the diagram of 4.9 along $\varphi_{-}$ to obtain the
sequence, noting that
$\varphi_{-,*}\mathcal{O}_{S}\cong\tilde{\varphi}_{+,*}\mathcal{O}_{S^{\nu}}\cong\mathcal{O}_{C}$
by 3.10, and that $\mathbf{R}^{1}\tilde{\varphi}_{+,*}\mathcal{O}_{S^{\nu}}=0$
since $\tilde{\varphi}_{+}$ is a projective bundle over $C$. Observe that
$\mathcal{F}$ is locally free since it is an extension of the locally free
$\mathcal{O}_{C}$-modules $\mathbf{R}^{1}\varphi_{-,*}\mathcal{O}_{S}$ is and
$\phi_{C,*}\mathcal{O}_{C}/\mathcal{O}_{C}$: the former is locally free by
3.10; for the latter, this follows by Kunz’s Theorem [Kun69, Theorem 2.1] or
[Stacks, 0EC0], since $\phi_{C}$ is, up to an automorphism, the
$q^{2}$-Frobenius of the regular curve $C$ by 1.6. The $\mathbf{G}_{m}$-action
from 4.2 provides $\mathcal{D}$ and $\mathcal{D}^{\nu}$ with gradings such
that $\mathcal{O}_{C}$ and $\phi_{C,*}\mathcal{O}_{C}$, being the constant
functions, make up the degree $0$ components. Therefore the positively graded
components of $\mathcal{F}$ map isomorphically to
$\mathbf{R}^{1}\varphi_{-,*}\mathcal{O}_{S}$ and provide the desired
splitting. ∎

The importance of $\mathcal{F}$ is in the following relation with the
cohomology of the structure sheaf of $S$:

######  4.13 Proposition. —

The cohomology of $\mathcal{O}_{S}$ is given by

$\mathrm{H}^{i}(S,\mathcal{O}_{S})\cong\begin{dcases}\mathrm{H}^{0}(C,\mathcal{O}_{C})&\text{if}\;i=0,\\\
\mathrm{H}^{0}(C,\mathcal{F})&\text{if}\;i=1,\;\text{and}\\\
\mathrm{H}^{1}(C,\mathcal{F})/\mathrm{H}^{1}(C,\mathcal{O}_{C})&\text{if}\;i=2.\end{dcases}$

###### Proof.

Consider the cohomology sequence associated with the exact sequence

$0\to\mathcal{O}_{S}\to\nu_{*}\mathcal{O}_{S^{\nu}}\to\nu_{*}\mathcal{O}_{S^{\nu}}/\mathcal{O}_{S}\to
0.$

Note that $\varphi_{-,*}\mathcal{O}_{S}\cong\mathcal{O}_{C}$ by 3.10 and, for
each $i=0,1,2$,

$\mathrm{H}^{i}(S,\nu_{*}\mathcal{O}_{S^{\nu}}/\mathcal{O}_{S})\cong\mathrm{H}^{i}(C,\mathcal{F})\quad\text{and}\quad\mathrm{H}^{i}(S^{\nu},\mathcal{O}_{S^{\nu}})\cong\mathrm{H}^{i}(C,\mathcal{O}_{C})$

since $\nu_{*}\mathcal{O}_{S^{\nu}}/\mathcal{O}_{S}$ is supported on $D$ and
$D\to C$ is affine, and and the fact that $\tilde{\varphi}_{+}\colon
S^{\nu}\to C$ is a projective bundle. Thus
$\mathrm{H}^{0}(S,\mathcal{O}_{S})\cong\mathrm{H}^{0}(S^{\nu},\mathcal{O}_{S^{\nu}})\cong\mathrm{H}^{0}(C,\mathcal{O}_{C})$
and there is an exact sequence

$0\to\mathrm{H}^{0}(C,\mathcal{F})\xrightarrow{a}\mathrm{H}^{1}(S,\mathcal{O}_{S})\xrightarrow{b}\mathrm{H}^{1}(S,\nu_{*}\mathcal{O}_{S^{\nu}})\to\mathrm{H}^{1}(C,\mathcal{F})\to\mathrm{H}^{2}(S,\mathcal{O}_{S})\to
0.$

The result will follow upon verifying that
$b\colon\mathrm{H}^{1}(S,\mathcal{O}_{S})\to\mathrm{H}^{1}(S,\nu_{*}\mathcal{O}_{S^{\nu}})$
vanishes. Since $\varphi_{-,*}\mathcal{O}_{S}\cong\mathcal{O}_{C}$, the Leray
spectral sequence gives a short exact sequence

$0\to\mathrm{H}^{1}(C,\mathcal{O}_{C})\xrightarrow{c}\mathrm{H}^{1}(S,\mathcal{O}_{S})\xrightarrow{d}\mathrm{H}^{0}(C,\mathbf{R}^{1}\varphi_{-,*}\mathcal{O}_{S})\to
0.$

The splitting in 4.12 implies that the composite $d\circ
a\colon\mathrm{H}^{0}(C,\mathcal{F})\to\mathrm{H}^{0}(C,\mathbf{R}^{1}\varphi_{-,*}\mathcal{O}_{S})$
is a surjection. So exactness of the long sequence means it remains to show
that $b\circ
c\colon\mathrm{H}^{1}(C,\mathcal{O}_{C})\to\mathrm{H}^{1}(S,\nu_{*}\mathcal{O}_{S^{\nu}})$
vanishes. Pushing down to $C$ along $\varphi_{-}$ and applying 4.8 shows that
this is

$\phi_{C}\colon\mathrm{H}^{1}(C,\mathcal{O}_{C})\to\mathrm{H}^{1}(C,\phi_{C,*}\mathcal{O}_{C})$

which, as in 4.12, is the map induced by the $q^{2}$-power Frobenius up to an
automorphism, and this is the zero map by 1.2. ∎

Structure sheaf cohomology of $S$ is therefore reduced to that of
$\mathcal{F}$, which will be computed when $q=p$ in 6.12. Combined with the
Euler characteristic computation in 1.13, this gives Theorem B. In turn,
cohomology of $\mathcal{F}$ is determined by describing its structure, and
this is achieved in the next section via the following duality relationship
with the algebra $\mathcal{D}$:

######  4.14 Proposition. —

There is a canonical isomorphism of graded $\mathcal{O}_{C}$-modules

$\mathcal{F}\cong\mathcal{D}^{\vee}\otimes\mathcal{O}_{C}(-q+1)\otimes
L_{+}^{\otimes 2q-1}\otimes L_{-}^{\otimes 2}.$

###### Proof.

Applying
$\mathbf{R}\mathcal{H}\\!\mathit{om}_{\mathcal{O}_{S}}(-,\mathcal{O}_{S})$ to
the ideal sheaf sequence of the conductor $D\hookrightarrow S$ yields a
triangle in the derived category of $S$

$\mathbf{R}\mathcal{H}\\!\mathit{om}_{\mathcal{O}_{S}}(\mathcal{O}_{D},\mathcal{O}_{S})\to\mathcal{O}_{S}\to\nu_{*}\mathcal{O}_{S^{\nu}}\xrightarrow{+1}$

since
$\mathbf{R}\mathcal{H}\\!\mathit{om}_{\mathcal{O}_{S}}(\mathcal{O}_{S},\mathcal{O}_{S})=\mathcal{O}_{S}$
and
$\mathbf{R}\mathcal{H}\\!\mathit{om}_{\mathcal{O}_{S}}(\operatorname{cond}_{\nu,S},\mathcal{O}_{S})=\nu_{*}\mathcal{O}_{S^{\nu}}$,
upon identifying $\operatorname{cond}_{\nu,S}$ with
$\nu_{*}\omega_{S^{\nu}/S}$ as in 4.9 and using duality for $\nu$. The map
$\mathcal{O}_{S}\to\nu_{*}\mathcal{O}_{S^{\nu}}$ is dual to evaluation at $1$,
and hence is the $\mathcal{O}_{S}$-module map determined by $1\mapsto 1$; in
other words, this is the map $\nu^{\\#}$, and so

$\nu_{*}\mathcal{O}_{S^{\nu}}/\mathcal{O}_{S}\cong\mathbf{R}\mathcal{H}\\!\mathit{om}_{\mathcal{O}_{S}}(\mathcal{O}_{D},\mathcal{O}_{S})[1]$

in the derived category of $S$. Applying $\mathbf{R}\varphi_{-,*}$ and
applying relative duality for $\varphi_{-}\colon S\to C$ yields

$\mathcal{F}\cong\mathbf{R}\varphi_{-,*}\mathbf{R}\mathcal{H}\\!\mathit{om}_{\mathcal{O}_{S}}(\mathcal{O}_{D},\mathcal{O}_{S})[1]\cong\mathbf{R}\mathcal{H}\\!\mathit{om}_{\mathcal{O}_{C}}(\mathbf{R}\varphi_{-,*}(\mathcal{O}_{D}\otimes\omega_{\varphi_{-}}),\mathcal{O}_{C})[1]$

where
$\omega_{\varphi_{-}}=(\omega_{S}\otimes\varphi_{-}^{*}\omega_{C}^{\vee})[1]$.
By 1.8,

$\mathcal{O}_{D}\otimes\omega_{S}\cong\mathcal{O}_{S}(2q-3)\rvert_{D}\otimes
L_{+}^{\vee,\otimes 2}\otimes L_{-}^{\vee,\otimes
2}\cong\varphi_{-}^{*}\mathcal{O}_{C}(2q-3)\rvert_{D}\otimes
L_{+}^{\vee,\otimes 2q-1}\otimes L_{-}^{\vee,\otimes 2}$

where
$\mathcal{O}_{S}(1)\rvert_{D}=\varphi_{-}^{*}\mathcal{O}_{C}(1)\rvert_{D}\otimes
L_{+}^{\vee}$ since $\varphi_{-}\colon D\to C$ is induced by the line
subbundle of $\mathcal{S}\rvert_{D}$ obtained by intersecting with
$(L_{-}\oplus W)_{D}$ by 2.5, so there is a short exact sequence

$0\to\varphi_{-}^{*}\mathcal{O}_{C}(-1)\rvert_{D}\to\mathcal{S}\rvert_{D}\to
L_{+,D}\to 0,$

and taking determinants yields the desired identification. Combining with
$\omega_{C}\cong\mathcal{O}_{C}(q-2)$ gives

$\mathbf{R}\varphi_{-,*}(\mathcal{O}_{D}\otimes\omega_{\varphi_{-}})=(\mathbf{R}\varphi_{-,*}\mathcal{O}_{D})\otimes\mathcal{O}_{C}(q-1)\otimes
L_{+}^{\vee,\otimes 2q-1}\otimes L_{-}^{\vee,\otimes 2}[1].$

Since $D\to C$ is of relative dimension $0$,
$\mathbf{R}\varphi_{-,*}\mathcal{O}_{D}=\varphi_{-,*}\mathcal{O}_{D}=\mathcal{D}$,
yielding the result. ∎

###  4.15. The algebra $\mathcal{D}^{\nu}$

— Before turning to $\mathcal{D}$, consider the simpler algebra
$\mathcal{D}^{\nu}$ associated with $\phi_{C}\circ\tilde{\varphi}_{+}\colon
D^{\nu}\to C$. Since $D^{\nu}$ is disjoint from the curve
$C_{-}^{\nu}=\mathbf{P}L_{-,C}$ parameterizing lines through $x_{-}\in X$, it
is contained in the $\mathbf{A}^{1}$-bundle over $C$ given by

$S^{\nu,\circ}\coloneqq S^{\nu}\setminus
C_{-}^{\nu}=\mathbf{P}(\mathcal{T}_{C}(-1)\oplus
L_{-,C})\setminus\mathbf{P}L_{-,C}\cong\mathbf{A}(\Omega_{C}^{1}(1)\otimes
L_{-}),$

the identification obtained by taking graphs of a linear function
$\mathcal{T}_{C}(-1)\to L_{-,C}$. Then 4.10 means that $D^{\nu}$ is the
$\delta$-order neighbourhood of the zero section $C_{+}^{\nu}$, so this
identifies the graded $\mathcal{O}_{C}$-module underlying the algebra of
$D^{\nu}$ as

$\mathcal{D}^{\nu}=(\phi_{C}\circ\tilde{\varphi}_{+})*\mathcal{O}_{D^{\nu}}\cong\phi_{C,*}\big{(}\bigoplus\nolimits_{i=0}^{\delta}(\mathcal{T}_{C}(-1)\otimes
L_{-}^{\vee})^{\otimes i}\big{)}.$

##  5\. Structure of $\mathcal{D}$ and $\mathcal{F}$

The purpose of this section is to describe the structure of the sheaves
$\mathcal{D}$ and—especially!—$\mathcal{F}$ in terms of more familiar sheaves
on the curve $C$. To give the main statement regarding $\mathcal{F}$, some
notation: View the restricted Euler sequence

$0\to\Omega_{\mathbf{P}W}^{1}\rvert_{C}\to
W^{\vee}\otimes\mathcal{O}_{C}(-1)\to\mathcal{O}_{C}\to 0$

as a $2$-step filtration on $W^{\vee}\otimes\mathcal{O}_{C}(-1)$, with the
sheaf of differentials the $0$-th filtered piece. This induces a $(d+1)$-step
filtration on the symmetric powers
$\operatorname{Sym}^{d}(W^{\vee})\otimes\mathcal{O}_{C}(-d)$ with graded
pieces

$\operatorname{gr}_{i}\big{(}\operatorname{Sym}^{d}(W^{\vee})\otimes\mathcal{O}_{C}(-d)\big{)}\cong\operatorname{Sym}^{d-i}(\Omega_{\mathbf{P}W}^{1}\rvert_{C})\;\;\text{for}\;0\leq
i\leq d.$

Generally, given any locally free sheaf $\mathcal{E}$ and $d\geq q$,
$\operatorname{Sym}^{d}(\mathcal{E})$ contains a subsheaf
$\mathcal{E}^{[1]}\otimes\operatorname{Sym}^{d-q}(\mathcal{E})$ consisting of
products of $q$-powers and monomials of degree $d-q$. Write

$\operatorname{Sym}^{d}_{\mathrm{red}}(\mathcal{E})\coloneqq\operatorname{Sym}^{d}(\mathcal{E})/(\mathcal{E}^{[1]}\otimes\operatorname{Sym}^{d-q}(\mathcal{E}))$

for the quotient by this subsheaf, with the convention that
$\operatorname{Sym}^{d}_{\mathrm{red}}=\operatorname{Sym}^{d}$ when $d<q$.
Finally, write
$\operatorname{Div}^{d}_{\mathrm{red}}(\mathcal{E})\coloneqq\operatorname{Sym}^{d}_{\mathrm{red}}(\mathcal{E}^{\vee})^{\vee}$
for the reduced $d$-th divided power of $\mathcal{E}$. The result is:

######  5.1 Proposition. —

The graded $\mathcal{O}_{C}$-module $\mathcal{F}$ carries a $q$-step
filtration such that

$\operatorname{Fil}_{0}\mathcal{F}\cong\bigoplus\nolimits_{b=0}^{q-2}\bigoplus\nolimits_{a=0}^{q-1}\operatorname{Div}^{q-2-b}(\mathcal{T})\otimes\mathcal{O}_{C}(-a)\otimes
L_{+}^{\otimes b}\otimes L_{-}^{\vee,\otimes a}.$

For each $0\leq b\leq 2q-3$, there is a canonical short exact sequence of
filtered bundles

$0\to\operatorname{Div}^{2q-3-b}_{\mathrm{red}}(W)\otimes\mathcal{O}_{C}\to\mathcal{F}_{bq+q-1}\to\operatorname{Div}^{q-3-b}(W)\otimes\mathcal{O}_{C}(q-1)\to
0.$

There is a degree $-q-1$ map $\partial\colon\mathcal{F}\to\mathcal{F}$ such
that, for each $0\leq i\leq q-1$ and $0\leq d\leq\delta-q-1$,
$\partial(\operatorname{Fil}_{i}\mathcal{F})\subseteq\operatorname{Fil}_{i-1}\mathcal{F}$
and
$\operatorname{gr}_{i}\partial\colon\operatorname{Fil}_{i}\mathcal{F}_{d+q+1}\to\operatorname{Fil}_{i-1}\mathcal{F}_{d}$
is an isomorphism if $p\nmid i$ and zero otherwise.

This is proved in 5.13 at the end of the section.

###  5.2. Affine bundles

— The duality relation 4.14 relates $\mathcal{F}$ with $\mathcal{D}$, and the
latter is a quotient of coordinate rings of schemes affine over $C$: namely,
$S^{\circ}\coloneqq S\setminus C_{-}$ and $T^{\circ}$ as in 2.7 with respect
to the smooth cone situation $(X,x_{-},\mathbf{P}L_{-}^{[1],\perp})$. These
lie in affine space bundles

$\mathbf{A}_{1}\coloneqq\mathbf{P}\mathcal{V}_{1}\setminus\mathbf{P}L_{-,C},\qquad\mathbf{A}_{2}\coloneqq\mathbf{P}\mathcal{V}_{2}\setminus\mathbf{P}\mathcal{T},\qquad\mathbf{B}\coloneqq\mathbf{P}\mathcal{V}\setminus\mathbf{P}(\mathcal{T}_{\pi_{1}}(-1,0))\rvert_{\mathbf{A}}$

with $\mathcal{V}_{1}$ and $\mathcal{V}_{2}$ as in 2.6, and $\mathcal{V}$ is
as in 2.11; set $\mathbf{A}\coloneqq\mathbf{A}_{1}\times_{C}\mathbf{A}_{2}$.
Comparing the diagram in 2.6 with the description of the boundaries
$T\setminus T^{\circ}$ from 3.1 implies that there is a commutative diagram of
affine schemes over $C$ given by

${D}$${S^{\circ}}$${\mathbf{B}}$${T^{\circ}}$${\mathbf{A}}$${C\makebox[0.0pt][l]{\,.}}$$\scriptstyle{\rho}$$\scriptstyle{\varphi}$$\scriptstyle{\pi}$

Observe that the relative Euler sequences for these affine bundles give
canonical isomorphisms

$\displaystyle\mathcal{O}_{\mathbf{A}}(-1,0)$
$\displaystyle\cong\pi^{*}\mathcal{O}_{C}(-1),$
$\displaystyle\mathcal{T}_{\pi_{2}}(0,-1)\rvert_{\mathbf{A}}$
$\displaystyle\cong\pi^{*}\mathcal{T},$ $\displaystyle\mathcal{O}_{\rho}(-1)$
$\displaystyle\cong\rho^{*}\mathcal{O}_{\mathbf{A}}(0,-1)\cong
L_{+,\mathbf{B}},$
$\displaystyle\mathcal{T}_{\pi_{1}}(-1,0)\rvert_{\mathbf{A}}$
$\displaystyle\cong L_{-,\mathbf{A}},$
$\displaystyle\mathcal{O}_{\mathbf{A}}(0,-1)$ $\displaystyle\cong
L_{+,\mathbf{A}},$ $\displaystyle\mathcal{T}_{\rho}(-1)\rvert_{\mathbf{B}}$
$\displaystyle\cong\rho^{*}\mathcal{T}_{\pi_{1}}(-1,0)\rvert_{\mathbf{B}}\cong
L_{-,\mathbf{B}}.$

This identifies the $\mathcal{O}_{C}$-algebras
$\mathcal{A}\coloneqq\pi_{*}\mathcal{O}_{\mathbf{A}}$ and
$\mathcal{B}\coloneqq\varphi_{*}\mathcal{O}_{\mathbf{B}}$ as follows:

######  5.3 Lemma. —

The $q$-bic form $\beta$ induces an isomorphism

$\mathcal{A}\cong\operatorname{Sym}^{*}(\mathcal{O}_{C}(-1)\otimes
L_{-}^{\vee})\otimes\operatorname{Sym}^{*}(\Omega_{\mathbf{P}W}^{1}(1)\rvert_{C}\otimes
L_{+}),$

and endows the $\mathcal{A}$-algebra $\mathcal{B}$ with an increasing
filtration whose associated graded pieces are

$\operatorname{gr}_{i}\mathcal{B}\coloneqq\operatorname{Fil}_{i}\mathcal{B}/\operatorname{Fil}_{i-1}\mathcal{B}\cong\mathcal{A}\otimes(L_{-}^{\vee}\otimes
L_{+})^{\otimes i}\quad\text{for all}\;i\in\mathbf{Z}_{\geq 0}.$

###### Proof.

The splittings $\mathcal{V}_{1}\cong\mathcal{O}_{C}(-1)\oplus L_{-,C}$ and
$\mathcal{V}_{2}\cong\mathcal{T}\oplus L_{+,C}$ as in §3 give canonical
relative projective coordinates on $\mathbf{P}\mathcal{V}_{i}$ over $C$, and
identifies the associated affine bundles as

$\mathbf{A}\cong\mathbf{A}(\mathcal{O}_{C}(1)\otimes
L_{-})\times_{C}\mathbf{A}(\mathcal{T}\otimes L_{+}^{\vee}).$

This identifies $\mathcal{A}=\pi_{*}\mathcal{O}_{\mathbf{A}}$ as claimed.

For $\mathcal{B}$, begin with the affine bundle
$\rho\colon\mathbf{B}\to\mathbf{A}$ obtained as the complement of
$\mathbf{P}(\mathcal{T}_{\pi_{1}}(-1,0))$ in $\mathbf{P}\mathcal{V}$ over
$\mathbf{A}$. Dualizing the short exact sequence in 2.13 and restricting to
$\mathbf{A}$ gives a sequence

$0\to\mathcal{O}_{\mathbf{A}}(0,1)\to\mathcal{V}^{\vee}\rvert_{\mathbf{A}}\to\Omega^{1}_{\pi_{1}}(1,0)\rvert_{\mathbf{A}}\to
0.$

View $\mathcal{V}^{\vee}\rvert_{\mathbf{A}}$ as the linear functions on
$\mathbf{P}\mathcal{V}\rvert_{\mathbf{A}}$ over $\mathcal{A}$. A local
generator for the subbundle $\mathcal{O}_{\mathbf{A}}(0,1)$ is then a linear
equation defining
$\mathbf{P}(\mathcal{T}_{\pi_{1}}(-1,0)\rvert_{\mathbf{A}})$, and so becomes
invertible on $\mathbf{B}$. Therefore

$\rho_{*}\mathcal{O}_{\mathbf{B}}\cong\operatorname{colim}_{n}\operatorname{Sym}^{n}(\mathcal{V}^{\vee}(0,-1))\rvert_{\mathbf{A}}$

where the transition maps are induced by multiplication by a local generator
for the subbundle
$\mathcal{O}_{\mathbf{A}}\hookrightarrow\mathcal{V}^{\vee}(0,-1)\rvert_{\mathbf{A}}$.
The $2$-step filtration on $\mathcal{V}^{\vee}(0,-1)\rvert_{\mathbf{A}}$
starting with $\mathcal{O}_{\mathbf{A}}$ and followed by the entire bundle
induces filtrations on the symmetric powers, compatible with the transition
maps, whence a filtration on $\rho_{*}\mathcal{O}_{\mathbf{B}}$ with graded
pieces

$\operatorname{gr}_{i}\rho_{*}\mathcal{O}_{\mathbf{B}}\cong\Omega^{1}_{\pi_{1}}(1,-1)\rvert_{\mathbf{A}}^{\otimes
i}\cong\mathcal{O}_{\mathbf{A}}\otimes(L_{-}^{\vee}\otimes L_{+})^{\otimes
i}\;\;\text{for all $i\in\mathbf{Z}_{\geq 0}$}$

upon applying the identifications of 5.2. Pushing along
$\pi\colon\mathbf{A}\to C$ then gives the result. ∎

###  5.4. Symmetries

— Consider the linear algebraic subgroup of $\operatorname{\mathbf{GL}}(V)$
which preserves the decomposition $V=W\oplus U$, fixes the point
$x_{-}=\mathbf{P}L_{-}$, and which acts via automorphisms on $C$:

$\operatorname{\mathbf{Aut}}(L_{-}\subset
U)\times\operatorname{\mathbf{Aut}}(W,\beta_{W})\cong\Set{\begin{pmatrix}\lambda^{-1}_{-}&\epsilon\\\
0&\lambda_{+}\end{pmatrix}\in\mathbf{GL}(L_{-}\oplus
L_{+})}\times\mathrm{U}_{3}(q).$

Via its natural linear action on $V$, it acts on each of the sheaves
$\mathcal{V}_{1}$, $\mathcal{V}_{2}$, and $\mathcal{V}$, and preserves the
subbundles excised in defining the tower of affine bundles
$\mathbf{B}\to\mathbf{A}\to C$. As such, the $\mathcal{O}_{C}$-algebras
$\mathcal{A}$ and $\mathcal{B}$ are equivariant for this group. Two pieces of
structure now leap to the forefront:

First, the action of the maximal torus $(\lambda_{-}^{-1},\lambda_{+})$ endows
$\mathcal{A}$ and $\mathcal{B}$ with a bigrading, normalized so that
$L_{-}^{\vee,\otimes a}\otimes L_{+}^{\otimes b}$ has weight
$(a,b)\in\mathbf{Z}_{\geq 0}^{2}$. The bigraded pieces are as follows:

######  5.5 Lemma. —

The form $\beta$ induces isomorphisms of filtered bundles

$\mathcal{B}_{(a,0)}\cong\mathcal{A}_{(a,0)}\cong\mathcal{O}_{C}(-a),\;\;\mathcal{B}_{(0,b)}\cong\mathcal{A}_{(0,b)}\cong\operatorname{Sym}^{b}(\Omega_{\mathbf{P}W}(1))\rvert_{C},\;\;\mathcal{B}_{(1,1)}\cong
W^{\vee}\otimes\mathcal{O}_{C}(-1),$

and
$\mathcal{B}_{(a,b)}\cong\operatorname{Fil}_{a}\big{(}\operatorname{Sym}^{b}(W^{\vee})\otimes\mathcal{O}_{C}(-a)\big{)}$
for all $a,b\in\mathbf{Z}_{\geq 0}$.

###### Proof.

Matching weights in 5.3 identifies the bigraded pieces of weights $(a,0)$ and
$(0,b)$ as claimed, and shows that the $(1,1)$ piece of $\mathcal{B}$ sits as
an extension

$0\to\Omega^{1}_{\mathbf{P}W}\rvert_{C}\to\mathcal{B}_{(1,1)}\to\mathcal{O}_{C}\to
0.$

Begin by considering the extension class of $\mathcal{V}$ in the sequence
2.13: By its construction in 2.11, the class $[\mathcal{V}]$ is the image
under the map, induced by pullback and pushforward along tautological maps,

$\operatorname{Ext}^{1}_{\mathbf{P}}(\pi^{*}\mathcal{V}_{2},\pi^{*}\mathcal{V}_{1})\to\operatorname{Ext}^{1}_{\mathbf{P}}(\mathcal{O}_{\mathbf{P}}(0,-1),\pi^{*}\mathcal{V}_{1})\to\operatorname{Ext}^{1}_{\mathbf{P}}(\mathcal{O}_{\mathbf{P}}(0,-1),\mathcal{T}_{\pi_{1}}(-1,0))$