# The Chow Ring Classes of $\operatorname{PGL}_{3}$ Orbit Closures in
$\mathbf{G}(1,5)$

Gaurav (Dhruv) Goel111Herchel Smith Harvard Undergraduate Summer Research
Program 2022

(October 2022)

###### Abstract

The space of all pencils of conics in the plane $\mathbf{P}V$ (where $\dim
V=3$) is a projective Grassmannian
$\mathbf{G}(1,\mathbf{P}\operatorname{Sym}^{2}V^{*})$ and admits a natural
$\operatorname{PGL}(V)$ action. It is a classical theorem that this action has
exactly eight orbits, and in fact that the orbit of a pencil
$\ell\subset\mathbf{P}\operatorname{Sym}^{2}V^{*}$ is determined completely by
its position with respect to the Veronese surface
$X\subset\mathbf{P}\operatorname{Sym}^{2}V^{*}$ of rank 1 conics and its
secant variety $S(X)\subset\mathbf{P}\operatorname{Sym}^{2}V^{*}$, which is
the cubic fourfold of rank 2 conics. In this paper, we present some geometric
descriptions of these orbits. Then, using a mixture of direct enumerative
techniques and some Chern class computations, we present a calculation of the
classes of the orbit closures in the Chow ring of this Grassmannian (and
consequently also of their degrees under the Plücker embedding
$\mathbf{G}(1,\mathbf{P}\operatorname{Sym}^{2}V^{*})\hookrightarrow\mathbf{P}\Lambda^{2}\operatorname{Sym}^{2}V^{*}$).

###### Contents

  1. 1 Introduction
  2. 2 Set-Up
    1. 2.1 Notation and Conventions
    2. 2.2 Classification of $\operatorname{PGL}_{3}$-orbits and Hasse Diagram
  3. 3 Calculations
    1. 3.1 Orbit $\mathcal{O}_{1}$
    2. 3.2 Orbit $\mathcal{O}_{2}$
    3. 3.3 Orbit $\mathcal{O}_{3}$
    4. 3.4 Orbit $\mathcal{O}_{4}$
    5. 3.5 Orbit $\mathcal{O}_{5}$
    6. 3.6 Orbit $\mathcal{O}_{6}$
    7. 3.7 Orbit $\mathcal{O}_{7}$
    8. 3.8 Orbit $\mathcal{O}_{8}$
  4. 4 Some Chern Class Computations
    1. 4.1 The Chow Ring Class of the Fano Scheme $F_{1}(S)$
    2. 4.2 Chern Classes of Bundles of Relative Principal Parts
  5. 5 Dealing with Characterstic $p>2$
  6. 6 Suggestions for Further Work

##  1 Introduction

Let $V$ be an $n+1$ dimensional vector space. The complete linear system

$\mathbf{P}H^{0}\mathcal{O}_{\mathbf{P}V}(2)\cong\mathbf{P}\operatorname{Sym}^{2}V^{*}\cong\mathbf{P}^{n(n+3)/2}$

of quadrics on $\mathbf{P}V=\mathbf{P}^{n}$ admits a natural action by
$\operatorname{PGL}(V)=\operatorname{PGL}_{n+1}$. It is a standard result from
linear algebra that this $\operatorname{PGL}(V)$ action has exactly $n+1$
orbits, classified by the rank of the corresponding bilinear form (at least
when the base field is algebraically closed). As natural generalization of
this question, we can ask the same question about families of quadrics in
$\mathbf{P}V$. The space of all $d$-dimensional linear systems of quadrics on
$\mathbf{P}V$ is the projective Grassmanian
$\mathbf{G}(d,\mathbf{P}\operatorname{Sym}^{2}V^{*})$, which consequently also
carries a natural $\operatorname{PGL}(V)$ action (the above being the case
$d=0$). The orbits of this $\operatorname{PGL}(V)$ action are irreducible
locally closed subsets of
$\mathbf{G}(d,\mathbf{P}\operatorname{Sym}^{2}V^{*})$, and we can ask many
questions about them, e.g.: What are their dimensions? Are there only finitely
many? If so, what is the corresponding orbit class space or its Hasse diagram
(see [1])? Where is each orbit closure smooth or singular? What are the
classes of the orbit closures in the Chow ring
$A\mathbf{G}(d,\mathbf{P}\operatorname{Sym}^{2}V^{*})?$

When $n=1$, we’re looking at the $\mathbf{P}^{2}$ of quadratic polynomials on
$\mathbf{P}^{1}$, and we have $d\in\\{0,1,2\\}$. For $d=0$, we have two
orbits: the dense orbit (corresponding to two distinct points) and the
discriminant conic (corresponding to double points); the
$\operatorname{PGL}(V)=\operatorname{PGL}_{2}$ in this case is exactly the
subgroup of $\operatorname{Aut}\mathbf{P}^{2}=\operatorname{PGL}_{3}$
preserving the discriminant conic. For $d=1,$ the situation is entirely dual:
we have the dense orbit (of lines transverse to the discriminant conic) and
the dual to the discriminant conic. Obviously, the case $d=2$ is trivial.

For $n\geq 2$, the simplest nontrivial case is $d=1$, corresponding to pencils
of quadrics. This corresponds to a $\operatorname{PGL}_{n+1}$ action on
$\mathbf{G}(1,n(n+3)/2)$. In this case, the classification is nontrivial but
well understood, at least in the case of pencils that contain at least one
smooth quadric. Here there are two invariants: a discrete invariant called the
Segre symbol and a continuous invariant corresponding to the multiset of $n+1$
points in $\mathbf{P}^{1}$ defined by the vanishing of the discriminant of the
pencil. It is a classical theorem of Weierstrass and Segre that these are the
only invariants (for a proof, see this historical account [2] or the classic
textbook [3]). The result of Weierstrass-Segre can also be interpreted very
geometrically as being a classification obtained by considering the positions
of the lines with respect to the various $\operatorname{PGL}_{n+1}$ orbit
closures in $\mathbf{P}^{n(n+3)/2}$ and the types of singularities of the base
locus of the pencil (see [4]).

Since multisets of $4$ or more points have continuous invariants222For
instance, for $4$ distinct points on $\mathbf{P}^{1}$, we have the cross ratio
$\lambda$, or more precisely the corresponding $j$-invariant
$j(\lambda)=256\frac{(1-\lambda+\lambda^{2})^{3}}{\lambda^{2}(1-\lambda)^{2}}.$
, the largest $n$ for which we can expect to have finitely many orbits is
$n=2$, i.e. the case of planar conics. The classification of pencils of plane
conics (i.e. the case $n=2,d=1$) was first obtained over $\mathbf{R}$ and
$\mathbf{C}$ by Jordan in 1906 (see [5], [6], or see [7, Ch. 16] if you prefer
English)333By contrast, the case $n=d=2$ (i.e. nets of plane conics) was
investigated only recently in 2021 by Abdallah, Emsalem, and Iarobbino in [8].
Here there is no dense orbit, but rather a 1-parameter family of dimension 8
orbits in $\mathbf{G}(2,5)$, along with 14 other orbits. This family is
parametrized by the $j$-invariant of the smooth plane cubic that is the locus
of singular conics in that net.. There are exactly 8 orbits in
$\mathbf{G}(1,5)$, which we call $\mathcal{O}_{i}$ for $i=1,\dots,8$. There
are five orbits,
$\mathcal{O}_{1},\mathcal{O}_{2},\mathcal{O}_{3},\mathcal{O}_{4},$ and
$\mathcal{O}_{5}$, of pencils containing at least one smooth conic,
corresponding to the general pencils, simply tangent pencils, bitangent
pencils, osculating pencils, and superosculating pencils respectively (c.f.
[7, §16.5] types I-V) and three types
$\mathcal{O}_{6},\mathcal{O}_{7},\mathcal{O}_{8}$ of singular pencils (i.e.
pencils containing only singular conics) described in detail in §2.2. The
Hasse diagram of their inclusions (i.e. a poset of the $\mathcal{O}_{i}$ with
an arrow $\mathcal{O}_{i}\to\mathcal{O}_{j}$ iff $\mathcal{O}_{j}$ is open in
$\overline{\mathcal{O}}_{i}\smallsetminus\mathcal{O}_{i}$) is given in Figure
1.

${\mathcal{O}_{3}}$${\mathcal{O}_{6}}$${\mathcal{O}_{1}}$${\mathcal{O}_{2}}$${\mathcal{O}_{5}}$${\mathcal{O}_{8}}$${\mathcal{O}_{4}}$${\mathcal{O}_{7}}$
Figure 1: Hasse Diagram of the $\operatorname{PGL}_{3}$ action on
$\mathbf{G}(1,5)$.

Each of these orbits has very interesting algebraic geometry associated to it.
For instance:

  1. (a)

The ideals of
$\overline{\mathcal{O}}_{5},\overline{\mathcal{O}}_{6},\overline{\mathcal{O}}_{7}$
are generated by quadratic polynomials in the Plücker coordinates.

  2. (b)

The orbit $\mathcal{O}_{2}$ is birational to a $\mathbf{P}^{3}$ bundle over a
cubic fourfold, namely the smooth locus $S_{\mathrm{sm}}$ of the secant
variety $S$ to the Veronese surface.

  3. (c)

The orbit $\mathcal{O}_{3}$ is birational to a $\mathbf{P}^{4}$ bundle over
$\mathbf{P}^{2}$.

  4. (d)

The orbit $\mathcal{O}_{5}$ is birational to bundle over $\mathbf{P}^{2}$ with
fibers quadric threefolds.

  5. (e)

The closure $\overline{\mathcal{O}}_{5}$ is non-normal and singular along
$\overline{\mathcal{O}}_{6}$, which is isomorphic to the Hilbert scheme
$\operatorname{Hilb}^{2}\mathbf{P}V^{*}$ of pairs of lines in the plane.

  6. (f)

The closure $\overline{\mathcal{O}}_{7}$ is the Segre fourfold
$\mathbf{P}^{2}\times\mathbf{P}^{2*}$ and $\mathcal{O}_{8}$ inside it is the
universal line, and so $\mathcal{O}_{8}$ is a $\mathbf{P}^{1}$ bundle over
$\mathbf{P}^{2}$.

  7. (g)

The closures $\overline{\mathcal{O}}_{6}$ and $\overline{\mathcal{O}}_{7}$ are
the irreducible components of the Fano variety of the cubic fourfold $S$.

See [9] or[10, §2] for proofs of some of the facts above; most of these will
also be reproven in the course of our discussion, see §3. In this paper, we
compute the classes of the closures $\overline{\mathcal{O}}_{i}$ in the Chow
ring $A\mathbf{G}(1,5)$ and their degrees under the Plücker embedding
$\mathbf{G}(1,5)\hookrightarrow\mathbf{P}^{14}$.444The Plücker degrees of the
Schubert cycles are well-known and can be computed by the hook length formula
(see [11] Theorem 2.3.1). Since the Plücker degree of any other closed
subscheme of the Grassmannian is the linear combination of its Chow ring
coefficients obtained by plugging in the appropriate Plücker degrees of
Schubert cycles, the Chow ring class is a more refined quantity associated to
the orbits than its Plücker degree. Some of these degrees can be found in [2]
Example 5.2 and [9] Proposition 1.1, and it is a good sanity check that our
methods, although very different from theirs, give the same results. Our main
result is:

######  Theorem 1.0.1.

The classes in the Chow ring $A\mathbf{G}(1,5)$ of the closures of the orbits
$\mathcal{O}_{1},\dots,\mathcal{O}_{8}$ described above, and their degrees
under the Plücker embedding are summarized in Table 1.

Orbit | Base Locus Type | Codimension |  $A\mathbf{G}(1,5)$ Class of Closure | Plücker Degree
---|---|---|---|---
$\mathcal{O}_{1}$ | $(1,1,1,1)$ | 0 | $\sigma_{0}$ | 14
$\mathcal{O}_{2}$ | $(2,1,1)$ | 1 | $6\sigma_{1}$ | 84
$\mathcal{O}_{3}$ | $(2,2)$ | 2 | $4\sigma_{2}$ | 36
$\mathcal{O}_{4}$ | $(3,1)$ | 2 | $6\sigma_{2}+9\sigma_{1,1}$ | 99
$\mathcal{O}_{5}$ | $(4)$ | 3 | $4\sigma_{3}+8\sigma_{2,1}$ | 56
$\mathcal{O}_{6}$ | $\\{*\\}$ | 4 | $3\sigma_{3,1}+6\sigma_{2,2}$ | 21
$\mathcal{O}_{7}$ | $L\cup\\{*\\}:*\notin L$ | 4 | $6\sigma_{3,1}+3\sigma_{2,2}$ | 24
$\mathcal{O}_{8}$ | $L\cup\\{*\\}:*\in L$ | 5 | $6\sigma_{4,1}+6\sigma_{3,2}$ | 18
Table 1: Classes in $A\mathbf{G}(1,5)$ of the $\operatorname{PGL}_{3}$ orbit
closures $\overline{\mathcal{O}}_{i}$ for $i=1,\dots,8.$

The rest of this paper is organized as follows. In §2, we establish our
notation and conventions, and then recall the classification of these orbits
and show how this relates to the positions of these lines with respect to the
Veronese surface and its secant variety. In §3, we give direct enumerative
arguments to compute the Chow ring classes of these orbits. In §4, we give a
second proof of most of these results by computing some Chern classes. In §5,
we explain how to modify our arguments in the case of positive characteristic
$p>2$. Finally, in §6, we present several suggestions for future work in this
direction.

The author was supported by the Herchel Smith Harvard Undegraduate Research
Program Summer 2020, and would like to thank Professor Joe Harris for his
supervision and guidance.

##  2 Set-Up

###  2.1 Notation and Conventions

We work over a fixed algebraically closed field $K$ of characteristic 0.555For
all practical purposes, $K=\mathbf{C}$ (this is somewhat justified by the
Lefschetz principle). Most of our results can be extended to algebraically
closed fields of positive characteristic $p>2$ without much difficulty,
although proofs of results invoking Kleiman’s theorem would have to be
replaced by a careful analysis of generic transversality of the corresponding
cycles (see §5). The situation in characteristic 2 is genuinely quite wonky.
For starters, we cannot use the correspondence between plane conics and
symmetric bilinear forms. Further, there are other pathologies, e.g. any
smooth plane conic is a strange curve, i.e. there is a unique point, called
the strange point, common to all tangents to the conic. We will not pursue
characteristic 2 any further; see [12] for more on conics in characteristic 2,
and see the work of Bhosle, e.g. [13], for more on pencils of quadrics in
characteristic 2. Let $V$ be a three dimensional $K$-vector space and let
$\mathbf{P}^{2}=\mathbf{P}V$ be the corresponding projective plane with
coordinates $x,y,z$. By $\mathbf{P}^{5}$, we always denote the space
$\mathbf{P}\operatorname{Sym}^{2}V^{*}$ of conics on $\mathbf{P}^{2}$, and
give it coordinates $a,b,c,h,e,f$ (in that order) so that the universal conic

$\Xi=\\{(C,p):p\in C\\}\subset\mathbf{P}^{5}\times\mathbf{P}^{2}$

is cut out by the equation

$ax^{2}+2hxy+by^{2}+2exz+2fyz+cz^{2}=0.$

In other words, the symmetric bilinear form on $V$ corresponding to a conic
with coordinates $a,b,c,h,e,f$ is defined by the matrix

$\begin{bmatrix}a&h&e\\\ h&b&f\\\ e&f&c\end{bmatrix}.$

Let

$\Delta:=\det\begin{bmatrix}a&h&e\\\ h&b&f\\\ e&f&c\end{bmatrix}=abc+2hef-
af^{2}-be^{2}-ch^{2}\in K[a,b,c,h,e,f]$

be the determinant of this matrix and let
$S=\mathbf{V}(\Delta)\subset\mathbf{P}^{5}$ be the cubic fourfold of singular
conics. Since the partial derivatives of $\Delta$ are exactly the six distinct
$2\times 2$ minors of this matrix (upto scalars), it follows that the singular
locus $X=S_{\mathrm{sing}}$ is exactly the locus of rank 1 conics. It is easy
to see that $X$ is the Veronese surface (the image of the “squaring map”
$\mathbf{P}V^{*}\to\mathbf{P}\operatorname{Sym}^{2}V^{*}$), and that $S$ is
its secant (and tangent) variety $S=S(X)=\tan(X)$, i.e. the union of all
secant lines to $X$, which also happens to be the union of all tangent lines
to $X$ (c.f. [14, Ch. 11]).

We let $\mathbf{G}(1,5)$ denote the Grassmannian
$\mathbf{G}(1,\mathbf{P}\operatorname{Sym}^{2}V^{*})$ of pencils of conics.
For each $r\geq 0$, we denote by $\Phi_{r}$ the incidence correspondence

$\Phi_{r}:=\\{(\ell,p):m_{p}(\ell,S)\geq
r\\}\subseteq\mathbf{G}(1,5)\times\mathbf{P}^{5}.$

We let $\pi_{1}$ and $\pi_{2}$ denote the projections on $\mathbf{G}(1,5)$ and
$\mathbf{P}^{5}$ respectively:

${\Phi_{r}}$${\mathbf{G}(1,5)}$${\mathbf{P}^{5}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$

Each $\Phi_{r}$ is clearly a closed subset; it is the vanishing locus of a
certain global section $\tau_{\Delta}$ of a vector bundle $\mathcal{E}^{r}$ on
$\Phi$ (see §4.2). For $r=0$, this is all of
$\mathbf{G}(1,5)\times\mathbf{P}^{5}$. For $r\geq 1$, this is a nonempty
proper closed subset. Since $\deg S=3$, for $r\geq 4$ these loci all coincide
and are simply

$\Phi_{4}:=\\{(\ell,p):p\in\ell\subseteq
S\\}\subset\mathbf{G}(1,5)\times\mathbf{P}^{5}.$

Finally, for each ordered pair $(a,b)$ of integers $4\geq a\geq b\geq 0$ and
complete flag $\mathcal{V}$ in $\mathbf{P}^{5}$, we let
$\Sigma_{a,b}(\mathcal{V})\subset\mathbf{G}(1,5)$ denote the corresponding
Schubert cycle. When $\mathcal{V}$ is irrelevant, we simply omit it and write
$\Sigma_{a,b}$. We let $\sigma_{a,b}=:=[\Sigma_{a,b}]\in A\mathbf{G}(1,5)$.
Finally, we let $\sigma_{a}:=\sigma_{a,b}$ as usual. Finally, we present the
following helpful lemma:

######  Lemma 2.1.1.

Let $N\geq 1$ be any integer. Then for any ordered pair $(a,b)$ of integers
with $N-1\geq a\geq b\geq 0$, the Plücker degree of the Schubert cycle
$\Sigma_{a,b}$ is

$\deg\Sigma_{a,b}=\binom{2N-2-a-b}{N-1-b}\frac{a-b+1}{N-b}.$

###### Proof.

By the hook length formula ([11] Theorem 2.3.1), we have

$\deg\Sigma_{a,b}=\frac{(2N-2-a-b)!}{\left(\prod_{k=1}^{N-1-a}(N+1-b-k)\right)\left(\prod_{k=N-a}^{N-1-b}(N-b-k)\right)\left(\prod_{k=1}^{N-1-a}(N-a-k)\right)},$

which simplifies to the suggested quantity. ∎

###  2.2 Classification of $\operatorname{PGL}_{3}$-orbits and Hasse Diagram

In this section, we recall the classification of
$\operatorname{PGL}_{3}$-orbits, interpret them geometrically based on the
position of the line $\ell\subset\mathbf{P}^{5}$ with respect to $X$ and $S$,
and figure out the Hasse diagram of the orbit closures. First recall the
following definition:

######  Definition 2.2.1.

Given a pencil $\ell$ of conics in $\mathbf{P}^{2}$, its base locus is the
closed subscheme

$B_{\ell}:=\bigcap_{C\in\ell}C\subset\mathbf{P}^{2}.$

This is, of course, also the intersection of any two distinct members of the
pencil. Suppose first that $\ell$ contains a smooth conic, say $C_{0}$; in
this case, if $C_{1}$ is any other member of $\ell$, then the base locus
$B_{\ell}=C_{0}\cap C_{1}$ is a 0-dimensional subscheme of $\mathbf{P}^{4}$ of
length 4. The underlying reduced subscheme therefore is a union of at most 4
points. Up to $\operatorname{PGL}_{3}$-equivalence, there are exactly five
such subschemes, and they are determined entirely by the multiplicities of the
points, which must form a partition of 4, and hence be given by one of the
following tuples:

$(1,1,1,1),(2,1,1),(2,2),(3,1),(4).$

######  Definition 2.2.2.

Given a pencil $\ell$ of conics containing a smooth conic, we define the type
of $\ell$ to be the partition of 4 corresponding to the multiplicites of the
points in its base locus.

Clearly, all five of these types are possible (simply take any two conics
intersecting in such a subscheme), and the type of base locus is a
$\operatorname{PGL}_{3}$-invariant of the pencil. This is in fact the only
invariant of pencils containing a smooth member, and this gives us five orbits
of pencils not contained in $S$.

######  Theorem 2.2.3 (Algebraic Description I).

There are exactly five $\operatorname{PGL}_{3}$-orbits of pencils of conics
containing at least one nonsingular member. These are given by:

  1. (a)

The orbit $\mathcal{O}_{1}=\operatorname{PGL}_{3}\langle
x^{2}+y^{2},x^{2}+z^{2}\rangle$ with base locus of type $(1,1,1,1)$ (general
pencil).

  2. (b)

The orbit $\mathcal{O}_{2}=\operatorname{PGL}_{3}\langle
x^{2}+y^{2},xz\rangle$ with base locus of type $(2,1,1)$ (simply tangent
pencil).

  3. (c)

The orbit $\mathcal{O}_{3}=\operatorname{PGL}_{3}\langle x^{2},yz\rangle$ with
base locus of type $(2,2)$ (bitangent pencil).

  4. (d)

The orbit $\mathcal{O}_{4}=\operatorname{PGL}_{3}\langle x^{2}+yz,xz\rangle$
with base locus of type $(3,1)$ (osculating pencil).

  5. (e)

The orbit $\mathcal{O}_{5}=\operatorname{PGL}_{3}\langle
x^{2},y^{2}+xz\rangle$ with base locus of type $(4)$ (superosculating pencil).

For a proof, see [7] Proposition 16.5.1 or [15] §1.7. From this description of
the base locus, this portion of the Hasse diagram is immediate, see Figure 2.

${\mathcal{O}_{3}}$${\mathcal{O}_{1}}$${\mathcal{O}_{2}}$${\mathcal{O}_{5}}$${\mathcal{O}_{4}}$
Figure 2: Hasse Diagram of the orbits containing at least one smooth conic.

We can also this interpret this theorem as a classification via the position
of the line relative to the secant variety $S$ to the Veronese surface. Given
a line $\ell\subset\mathbf{P}^{5}$ not contained in $S$, let
$C,C^{\prime}\in\ell$ be two distinct elements and $Q,Q^{\prime}$ be the
corresponding symmetric $3\times 3$ matrices, so that the matrices
$sQ+tQ^{\prime}$ for $[s,t]\in\mathbf{P}^{1}$ give a parametrization of
$\ell$. The pullback of $\Delta$ to $\mathbf{P}^{1}$ is exactly
$\det(sQ+tQ^{\prime})$, which is a cubic polynomial in $s$ and $t$.666This
already proves the not-entirely-trivial fact that if a pencil contains at
least one smooth member, then it has at most three singular elements; this
corresponds to the geometric statement that a line $\ell$ not contained in $S$
meets it in at most $\deg S=3$ distinct points. The roots of this cubic
polynomial then tell us how $\ell$ interesects $S$. For instance, for type
$\mathcal{O}_{2}$, we can take $C=\mathbf{V}(x^{2}+y^{2})$ and
$C^{\prime}=\mathbf{V}(2xz)$ and so

$Q=\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&0\end{bmatrix}\text{ and
}Q^{\prime}=\begin{bmatrix}0&0&1\\\ 0&0&0\\\ 1&0&0\end{bmatrix}.$

Then

$sQ+tQ^{\prime}=\begin{bmatrix}s&0&t\\\ 0&s&0\\\
t&0&0\end{bmatrix}\Rightarrow\det(sQ+tQ^{\prime})=-st^{2}.$

This tells us that this $\ell$ intersects $S$ in two smooth points, once with
multiplicity 1 and another with multiplicity 2. Carrying out the analysis for
all five types above, we get a geometric characterization of the orbits:

######  Theorem 2.2.4 (Geometric Description I).

The five $\operatorname{PGL}_{3}$-orbits of lines $\ell\subset\mathbf{P}^{5}$
not contained in $S$ are characterized by their position relative to $S$,
namely:

  1. (a)

$\mathcal{O}_{1}$ is the orbit of lines intersecting $S$ transversely in three
smooth points,

  2. (b)

$\mathcal{O}_{2}$ is the orbit of lines intersecting $S$ in two smooth points,
once with multiplicity 2,

  3. (c)

$\mathcal{O}_{3}$ is the orbit of lines intersecting $S$ in one singular and
one smooth point,

  4. (d)

$\mathcal{O}_{4}$ is the orbit of lines intersecting $S$ in a unique smooth
point with multiplicity 3, and

  5. (e)

$\mathcal{O}_{5}$ is the orbit of lines intersecting $S$ in a unique singular
point with multiplicity 3.

Next, we look at the singular pencils (i.e. pencils consisting only of
singular members). In this case, the base locus can be positive dimensional.

######  Theorem 2.2.5 (Algebraic Description II).

There are exactly 3 $\operatorname{PGL}_{3}$-orbits of singular pencils of
conics. These are given by:

  1. (a)

The orbit $\mathcal{O}_{6}=\operatorname{PGL}_{3}\langle x^{2},y^{2}\rangle$,
the unique orbit of singular pencils of base loci consisting of a single
point.

  2. (b)

The orbit $\mathcal{O}_{7}=\operatorname{PGL}_{3}\langle xy,xz\rangle$, of
pencils with base locus consisting of a line $L$ and point $*\notin L$.

  3. (c)

The orbit $\mathcal{O}_{8}=\operatorname{PGL}_{3}\langle x^{2},xy\rangle$, of
pencils with base locus consisting of a line $L$ and an embedded point $*\in
L$.

A proof can be found in [15] §1.7. Doing the same analysis as before shows
that we can understand this theorem as a classification via the position of
the line relative to the Veronese surface $X$.

######  Theorem 2.2.6 (Geometric Description II).

The three $\operatorname{PGL}_{3}$-orbits of lines $\ell\subset\mathbf{P}^{5}$
contained in $S$ are characterized by their positions relative to $X$, namely:

  1. (a)

$\mathcal{O}_{6}$ is the orbit of lines intersecting $X$ in two distinct
points,

  2. (b)

$\mathcal{O}_{7}$ is the orbit of lines not intersecting $X$ at all, and

  3. (c)

$\mathcal{O}_{8}$ is the orbit of lines tangent to $X$.777A line cannot meet
the Veronese surface in a positive dimensional scheme or a zero-dimensional
scheme of length 3 or higher because the Veronese surface is cut out by
quadrics and contains no lines (because of degree reasons).

In this case, the specialization results are not entirely obvious. The
specialization $\mathcal{O}_{7}\to\mathcal{O}_{8}$ is easy to see from the
base locus (algebraic) description, whereas the specialization
$\mathcal{O}_{6}\to\mathcal{O}_{8}$ is easy to see from the geometric
description. We will see below that $\mathcal{O}_{6}$ and $\mathcal{O}_{7}$
are both 4-dimensional and so cannot be comparable for the specialization
partial order. Next, considering the family $\langle x^{2},y^{2}+txz\rangle$
for $t\in\mathbf{A}^{1}$ gives us the specialization
$\mathcal{O}_{5}\to\mathcal{O}_{6}$. Similarly, considering the family
$\langle xz,yz+tx^{2}\rangle$ for $t\in\mathbf{A}^{1}$ gives us the
specialization $\mathcal{O}_{4}\to\mathcal{O}_{7}$. What is also true but
perhaps less obvious is that $\mathcal{O}_{5}$ does not specialize to
$\mathcal{O}_{7}$; indeed, this is follows from the geometric description,
since interesecting the Veronese surface $X$ is a closed condition in the
Grassmannian. Putting all of this together, and combining it with the
dimension computations of the orbits in the next section, we obtain the Hasse
diagram, Fig. 1, of orbit inclusions; c.f. [9] §1.2 and [8] §3.2. Finally, we
also see that $\overline{\mathcal{O}}_{6}$ and $\overline{\mathcal{O}}_{7}$
are the two irreducible components of the Fano variety of $S$, as described in
[10, §2]. We will see in §3 that the Fano scheme of $S$ consists of
$\overline{\mathcal{O}}_{6}$ with multiplicity four and
$\overline{\mathcal{O}}_{7}$ with multiplicity one.

##  3 Calculations

###  3.1 Orbit $\mathcal{O}_{1}$

The orbit $\mathcal{O}_{1}$ is the orbit of pencils with base locus type
$(1,1,1,1)$ and corresponds to lines $\ell\subset\mathbf{P}^{5}$ meeting $S$
transversely in three smooth points. Therefore, $\mathcal{O}_{1}$ is a
nonempty open (and hence dense) orbit. Equivalently, if
$U\subset\operatorname{Hilb}^{4}\mathbf{P}^{2}$ is the open subset of points
in linear general position, then the map $U\to\mathbf{G}(1,5)$ sending a base
locus to the pencil of conics through those 4 points is an open embedding
whose image is exactly $\mathcal{O}_{1}$. The Plücker degree of its closure is
the Plücker degree of $\mathbf{G}(1,5)$ itself, namely 14, computed say using
Lemma 2.1.1 as $\deg\Sigma_{0,0}$.

###  3.2 Orbit $\mathcal{O}_{2}$

This is the orbit of pencils corresponding to lines simply tangent to $S$ at a
smooth point.

######  Proposition 3.2.1.

The closure $\overline{\mathcal{O}}_{2}$ is the variety $\mathcal{T}_{1}(S)$
of lines tangent to $S$. In particular, $\mathcal{O}_{2}$ is birational to a
$\mathbf{P}^{3}$-bundle over $S_{\mathrm{sm}}$, so $\dim\mathcal{O}_{2}=7$.

###### Proof.

Look at the incidence correspondence $\Phi_{2}$ defined in §2.1 above. The
projection $\pi_{2}$ is dominant onto $S$. Given any $p\in S_{\mathrm{sm}}$,
the fiber $\pi_{2}^{-1}(p)\subset\Phi_{2}$ is the set of lines tangent to $S$
at $p$, and these are exactly the lines through $p$ in
$\mathbf{T}_{p}S\cong\mathbf{P}^{4}\subset\mathbf{P}^{5}$.888In fact, suppose
we have a smooth point $p=[LM]\in S_{\mathrm{sm}}$ corresponding to the
product of two distinct lines $L,M\subset\mathbf{P}^{2}$ and another point
$q\in\mathbf{P}^{5}\smallsetminus S$ with corresponding conic
$C_{q}\subset\mathbf{P}^{2}$. Then the line $\ell=\overline{p,q}$ is tangent
to $S$ at $p$ iff $C_{q}\ni L\cap M$. In other words, the tangent hyperplane
$\mathbf{T}_{p}S$ is the hyperplane in $\mathbf{P}^{5}$ of conics in the plane
containing the point $L\cap M$.It follows that $\Phi_{2}$ is the closure of
projectivized geometric tangent bundle $\mathbf{P}TS_{\mathrm{sm}}$ over
$S_{\mathrm{sm}}$. Since $S$ is irreducible, it follows that $\Phi_{2}$ is
irreducible of dimension 7, and birational to a $\mathbf{P}^{3}$-bundle over
$S_{\mathrm{sm}}$.

Now $\pi_{1}(\Phi_{2})=:\mathcal{T}_{1}(S)$ is the variety of tangent lines to
$S$ (where we use the notation of [14, §15.7]; since $S$ is singular we need
to be a little careful in how we define this variety). In here,
$\mathcal{O}_{2}$ is an open subset, since it is the complement of the locus
of lines intersecting the Veronese surface or tangent to $S$ to order 3 or
higher. It follows that
$\overline{\mathcal{O}}_{2}=\pi_{1}(\Phi_{2})=\mathcal{T}_{1}(S)$. Finally,
the projection $\pi_{1}:\Phi_{2}\to\overline{\mathcal{O}}_{2}$ is regular and
birational, since a line simply tangent to the cubic hypersurface $S$ is
tangent to it at a unique point. ∎

######  Proposition 3.2.2.

The class $[\overline{\mathcal{O}}_{2}]=6\sigma_{1}$.

We give three proofs of this result, two here and one in §3.

###### Proof 1 of Proposition 3.2.2..

Write $[\overline{\mathcal{O}}_{2}]=a\sigma_{1}$ for some
$a\in\mathbf{Z}_{\geq 0}$. To figure $a$ out, we take the product on both
sides with $\sigma_{4,3}\in A\mathbf{G}(1,5)$. By Kleiman’s transversality
theorem, to calculate $a$, we’re asking the question: given a general 2-plane
$\Lambda\subset\mathbf{P}^{5}$ and a general point $p\in\Lambda$, how many
lines through $p$ and contained in $\Lambda$ are tangent to $S$? Since a
general $2$-plane $\Lambda\subset\mathbf{P}^{5}$ does not meet the two-
dimensional $X$, Bertini’s and Bezout’s theorems tell us that for a general
2-plane $\Lambda$ the intersection $\Lambda\cap S$ will be a smooth plane
cubic, and so given a general point $p\in\Lambda$, the number of lines through
it tangent to this smooth cubic curve will be exactly the degree of the dual
curve to a smooth cubic. It is well-known that the degree of the curve dual to
a smooth plane curve of degree $d$ is $d(d-1)$; for $d=3$, this is $a=6$. ∎

###### Proof 2 of Proposition 3.2.2.

To answer the same enumerative question differently, let $\Lambda$ be the span
of the three points $p,p^{\prime},p^{\prime\prime}$ corresponding to matrices
$Q,Q^{\prime},Q^{\prime\prime}$ respectively. The $\mathbf{P}^{1}$ of lines in
$\Lambda$ through $p$ then can be parametrized as $\langle Q,\lambda
Q^{\prime}+\mu Q^{\prime\prime}\rangle$ where
$[\lambda,\mu]\in\mathbf{P}^{1}$. Writing

$\det(sQ+t(\lambda Q^{\prime}+\mu
Q^{\prime\prime}))=f_{0}(\lambda,\mu)s^{3}+f_{1}(\lambda,\mu)s^{2}t+f_{2}(\lambda,\mu)st^{2}+f_{3}(\lambda,\mu)t^{3},$

where $f_{i}(\lambda,\mu)\in K[\lambda,\mu]$ is homogenous of degree $i$, we
see that this cubic in $s$ and $t$ has a double root iff the discriminant

$\displaystyle q_{6}(\lambda,\mu)$
$\displaystyle=\mathrm{disc}_{[s,t]}\det(sQ+t(\lambda Q^{\prime}+\mu
Q^{\prime\prime}))$
$\displaystyle=18f_{0}f_{1}f_{2}f_{3}-4f_{1}^{3}f_{3}+f_{1}^{2}f_{2}^{2}-4f_{0}f_{2}^{2}-27f_{0}^{2}f_{3}^{2}$

vanishes. This discriminant is a sextic polynomial and so generally has $a=6$
roots in $\mathbf{P}^{1}$. ∎

######  Corollary 3.2.3.

The Plücker degree $\deg\overline{\mathcal{O}}_{2}=84$.

###### Proof.

Taking $N=5,a=1,b=0$ in Lemma 2.1.1, we see $\deg\Sigma_{1}=14$, from which it
follows that $\deg\overline{\mathcal{O}}_{2}=6\deg\Sigma_{1}=84$. ∎

###  3.3 Orbit $\mathcal{O}_{3}$

This is the orbit of pencils that meet $S$ in two points, one of them being on
$X$.

######  Proposition 3.3.1.

The closure $\overline{\mathcal{O}}_{3}$ is the variety $\mathcal{C}_{1}(X)$
of lines incident to the Veronese surface. It is birational to a
$\mathbf{P}^{4}$ bundle over $\mathbf{P}^{2}$, and so irreducible of dimension
6.

###### Proof.

Consider the incidence correspondence

$\Psi:=\\{(\ell,p):p\in\ell\cap X\\}\subset\mathbf{G}(1,5)\times X$

and the projections $\pi_{1}$ and $\pi_{2}$ to $\mathbf{G}(1,5)$ and $X$
respectively. This is simply the projectivized tautological quotient bundle
restricted to $X$, i.e. $\Psi=\mathbf{P}\mathcal{Q}|_{X}$, and so $\Psi$ is
irreducible of dimension 6. On the other hand,
$\pi_{1}(\Psi)=:\mathcal{C}_{1}(X)$ and the projection $\pi_{1}$ is a regular
birational map because a general line incident to $X$ is incident to it at a
unique point999This follows from the existence of one such line, which is easy
to establish. An equivalent way of say thing is that the variety
$\mathcal{S}X\subset\mathbf{G}(1,5)$ of secant lines to $X$, which has
dimension 4, is a proper subvariety of $\mathcal{C}_{1}(X)$. It follows that
$\mathcal{C}_{1}(X)$ is also irreducible of dimension 6. Since
$\mathcal{O}_{3}$ is open in $\mathcal{C}_{1}(X)$ (it is the complement in
$\mathcal{C}_{1}(X)$ of lines meeting $S$ at a single point to order 3 or
more), it follows that $\overline{\mathcal{O}}_{3}=\mathcal{C}_{1}(X)$. ∎

######  Proposition 3.3.2.

The class $[\overline{\mathcal{O}}_{3}]=4\sigma_{2}$.

###### Proof.

We write $[\overline{\mathcal{O}}_{3}]=a\sigma_{2}+b\sigma_{1,1}$ for some
$a,b\in\mathbf{Z}_{\geq 0}$. To compute $a$ and $b$, we take the product of
$[\overline{\mathcal{O}}_{3}]$ with $\sigma_{4,2}$ and $\sigma_{3,3}$
respectively. Taking the product with $\sigma_{4,2}$ amounts to asking: given
a general 3-plane $\Pi\subset\mathbf{P}^{5}$ and general point
$p\in\mathbf{P}$, how many lines containing a $p$ and contained in $\Pi$
intersect the Veronese surface? Since the Veronese surface has dimension 2 and
degree 4, a general 3-plane $\Pi\subset\mathbf{P}^{5}$ meets the Veronese
surface in 4 distinct points in linear general position.101010To show that
these points are in linear general position, it suffices to observe that a
general hyperplane section of the Veronese surface is a rational normal curve
of degree 4. Since a general point $p\in\Pi$ will not lie on the
$\binom{4}{2}=6$ lines spanned by these points, it follows that there will be
exactly $a=4$ lines through $p$ contained in $\Pi$ and meeting the Veronese
surface.

Taking the product with $\sigma_{3,3}$ amounts to asking: given a general
2-plane $\Lambda\subset\mathbf{P}^{2}$, how many lines intersecting the
Veronese surface are contained in $\Lambda$? Well, a general 2-plane $\Lambda$
doesn’t intersect the Veronese surface at all (for dimensional reasons), and
so the answer is $b=0$. ∎

######  Corollary 3.3.3.

The Plücker degree $\deg\overline{\mathcal{O}}_{3}=36$.

###### Proof.

Combine the previous result with $\deg\Sigma_{2}=9$ obtained from Lemma 2.1.1.
∎

###  3.4 Orbit $\mathcal{O}_{4}$

This is the orbit of pencils corresponding to the lines meeting $S$ in a
single smooth point with multiplicity 3.

######  Proposition 3.4.1.

The orbit $\mathcal{O}_{4}$ is birational to a bundle over $S_{\mathrm{sm}}$
with each fiber consisting of two planes intersecting in a line and with
nontrivial monodromy. In particular, $\overline{\mathcal{O}}_{4}$ is
irreducible of dimension 6.

###### Proof.

Look at the incidence correspondence $\Phi_{3}$. On the one hand, again
$\pi_{2}(\Phi_{3})=S$. Given any $p=[LM]\in S_{\mathrm{sm}}$ corresponding to
the product of the lines $L,M\subset\mathbf{P}^{2}$ and another point
$q\in\mathbf{P}^{5}\smallsetminus S$ corresponding to the conic
$C_{q}\subset\mathbf{P}^{2}$, it is easy to see that the line
$\ell=\overline{p,q}$ spanned by $p$ and $q$ intersects $S$ with multiplicity
3 at $p$ iff $C_{q}$ is tangent to either $L$ or $M$ at $L\cap M$. For a fixed
$p$, the set of all such $q$ is the union of two 3-planes contained in
$\mathbf{T}_{p}S$ intersecting in the 2-plane of conics singular at $L\cap M$.
In particular, given any $p\in S_{\mathrm{sm}}$, the fiber
$\pi_{2}^{-1}(p)\subset\Phi_{3}$ is the projection of this locus from $p$, so
the union of two 2-planes intersecting in a line.

Further, this map $\pi_{2}:\Phi_{3}\to S$ has nontrivial monodromy (which is
equivalent to saying that if $\Phi_{3}\to\hat{S}\to S$ is the Stein
factorization of $\pi_{2}$, then the 2-sheeted cover $\hat{S}\to S$ is
nontrivial). Indeed, this follows from the fact that after removing the
$\mathbf{P}^{1}$-bundle given by the intersection of the two planes that make
up each fiber, the two irreducible components of the remaining fiber over a
$p=[LM]$ correspond to tangency to $L$ or $M$; the lift of a loop on $S$ that
corresponds to switching $L$ and $M$ then lifts to a path in the Stein
factorization $\hat{S}$ that connects the two points on the fiber. In
particular, it follows that $\Phi_{3}$ is irreducible of dimension 6.

Since $\mathcal{O}_{4}\subset\pi_{1}(\Phi_{3})$ is an open subset (it is the
complement of lines meeting the Veronese surface), it follows that
$\overline{\mathcal{O}}_{4}=\pi_{1}(\Phi_{3})$. Finally, the map
$\pi_{1}:\Phi_{3}\to\overline{\mathcal{O}}_{4}$ is regular and birational
because the fiber over any $\ell\in\mathcal{O}_{4}$ is its unique point of
intersection with $S$. ∎

######  Proposition 3.4.2.

The class $[\overline{\mathcal{O}}_{4}]=6\sigma_{2}+9\sigma_{1,1}$.

Again, we give two proofs of this result, one here and one in $\S 3$.

###### Proof 1 of Proposition 3.4.2.

Write $[\overline{\mathcal{O}}_{4}]=a\sigma_{2}+b\sigma_{1,1}$ and take
products with $\sigma_{4,2}$ and $\sigma_{3,3}$ as before.Taking the product
with $\sigma_{4,2}$ amounts to asking: given a general 3-plane
$\Pi\subset\mathbf{P}^{5}$ and a general point $p\in\mathbf{P}$, how many
lines containing $p$ and contained in $\Pi$ are tangent to the cubic fourfold
$S$ to order 3? If the intersection $S\cap\Pi$ were a smooth cubic surface,
then the answer would be classical: the projection of smooth cubic surface in
$\mathbf{P}^{3}$ from a general point expresses it as a 3-sheeted cover of
$\mathbf{P}^{2}$ branched along a sextic with six cusps, and so there would be
exactly $a=6$ such lines. The current problem is complicated by the fact
$S\cap\Pi$ is not smooth; however, it is not too singular either (e.g. it is
normal111111It is normal by Serre’s $R_{1}+S_{2}$ criterion because it is a
hypersurface and regular in codimension 1. and has finitely many isolated
singularities), and the answer remains the same. Indeed, by Bertini’s theorem,
a general 3-plane $\Pi$ intersects $S$ in a cubic surface singular at 4
distinct points (namely the four points of $X\cap\Pi$), and there is only one
such variety: the Cayley cubic, which is obtained by blowing up the plane
along the 6 vertices of a complete quadrilateral (see [16, pp. 640-646] ,
[17], or [18, §4.1.3]). General projections of cubic surfaces with ordinary
double points have been studied as well (see [19]); since we’ll give another
proof of this result below, we’ll content ourselves with quoting the results
of [19, §4.7] to conclude that $a=6$.

Finally, taking the product with $\sigma_{3,3}$ amounts to asking: given a
general 2-plane $\Lambda\subset\mathbf{P}^{5}$, how many lines contained in
$\Lambda$ are triply tangent to $S$? By Bertini’s and Bezout’s theorems, a
general $\Lambda$ intersects $S$ in a smooth cubic curve, and it is well-known
that a smooth cubic has exactly $b=9$ flexes. ∎

######  Corollary 3.4.3.

The Plücker degree $\deg\overline{\mathcal{O}}_{4}=99$.

###### Proof.

Combine the previous result with $\deg\Sigma_{2}=9$ and $\deg\Sigma_{1,1}=5$.
∎

###  3.5 Orbit $\mathcal{O}_{5}$

This the orbit of pencils that meet $S$ is a unique singular point (i.e. in a
point of $X$) with multiplicity 3.

######  Proposition 3.5.1.

The orbit $\mathcal{O}_{5}$ is birational to a bundle over $\mathbf{P}^{2}$
whose fiber is a rank 3 quadric threefold. In particular,
$\overline{\mathcal{O}}_{5}$ is irreducible of dimension 5.

###### Proof.

Set up the incidence correspondance

$\Theta:=\\{(\ell,p):p\in\ell\cap X\text{ and }m_{p}(S,\ell)\geq
3\\}\subset\mathbf{G}(1,5)\times X,$

and let $\pi_{1},\pi_{2}$ denotes projections to $\mathbf{G}(1,5)$ and $X$
respectively. Given a point $p=[L^{2}]\in X$ corresponding to the double line
$L^{2}\subset\mathbf{P}^{2}$ and another point
$q\in\mathbf{P}^{5}\smallsetminus S$ corresponding to the conic
$C_{q}\subset\mathbf{P}^{2}$, it is easy to see that the line
$\ell=\overline{p,q}$ joining $p$ and $q$ intersects $S$ with multiplicity 3
at $p$ iff $q$ lies in the tangent cone $\operatorname{TC}_{p}S$ to $S$ at
$p$, which happens iff the conic $C_{q}$ is tangent to the line $L$ somewhere.
The tangent cone $\operatorname{TC}_{p}S$ is a rank 3 quadric in
$\mathbf{P}^{5}$ containing $p$ in its vertex, and so for a fixed $p$, in the
$\mathbf{P}^{4}$ given by the projection of $\mathbf{P}^{5}$ from $p$, this
condition on $q$ defines a rank 3 quadric threefold
$\pi_{2}^{-1}(p)\subset\Theta$. It follows that $\Theta$ is irreducible of
dimension 5. Since $\mathcal{O}_{5}$ is a nonempty open subset of
$\pi_{2}(\Theta)$ (namely the complement in $\pi_{2}\Theta$ of the Fano
variety $F_{1}(S)\subset\mathbf{G}(1,5)$), it follows that
$\overline{\mathcal{O}}_{5}=\pi_{2}(\Theta)$. Finally, the map
$\pi_{2}:\Theta\to\overline{\mathcal{O}}_{5}$ is regular and birational
(because the fiber over any point in $\mathcal{O}_{5}$ is a single point).

∎

######  Proposition 3.5.2.

The class $[\overline{\mathcal{O}}_{5}]=4\sigma_{3}+8\sigma_{2,1}$.

###### Proof.

Let $[\overline{\mathcal{O}}_{5}]=a\sigma_{3}+b\sigma_{2,1}$. To figure $a$
out, we take the product with $\sigma_{4,1}$ and this amounts to asking: given
a general point $q\in\mathbf{P}^{5}$ and a general hyperplane
$H\subset\mathbf{P}^{5}$ containing $q$, how many lines through $q$ contained
in $H$ meet $S$ in a unique point of $X$? We take
$q\in\mathbf{P}^{5}\smallsetminus S$. If $C_{q}\subset\mathbf{P}^{2}$ is the
corresponding smooth conic, then the line joining $q$ and a point $[L^{2}]\in
X$ meets $S$ in $[L^{2}]$ only iff $L$ is tangent to $C_{q}$. In particular,
the locus of all such $L$ is the dual conic $C_{q}^{*}\subset\mathbf{P}^{2*}$,
which under the Veronese map $\mathbf{P}^{2*}\to X$ maps to a rational normal
quartic in $\mathbf{P}^{5}$. It follows that a general hyperplane $H$
containing $q$ interesects this curve in $a=4$ points.

To figure $b$ out, we take the product with $\sigma_{3,2}$, and this amounts
to asking: given a general line $\ell\subset\mathbf{P}^{5}$ and a general
$3$-plane $\Pi\subset\mathbf{P}^{5}$ containing $\ell$, how many lines
incident to $\ell$ and contained in $\Pi$ meet $S$ in a unique point of $X$?
By Bertini’s theorem, a general 3-plane section $\Pi\cap X$ of $X$ consists of
4 points in linear general position, say $p_{i}=[L_{i}^{2}]$ for $i=1,2,3,4$,
where the $L_{i}\subset\mathbf{P}^{2}$ are four lines in linear general
position. For each such point $p_{i}$, the tangent cone
$\operatorname{TC}_{p_{i}}S$ intersects the 3-plane $\Pi$ in a rank 3 quadric
surface, i.e. in a cone over a plane conic. No two of these quadric surfaces
coincide, and so the intersection of any of two of them has positive
codimension in either.121212In fact, the intersection
$\operatorname{TC}_{p_{i}}S\cap\operatorname{TC}_{p_{j}}S$ for $i\neq j$ is
the locus of conics that are tangent to both $L_{i}$ and $L_{j}$. This locus
has two components, namely the locus of conics through $L_{i}\cap L_{j}$ and
another component. The first of these is a hyperplane along which
$\operatorname{TC}_{p_{i}}S$ and $\operatorname{TC}_{p_{j}}S$ meet with
multiplicity two, and whose intersection with $\Pi$ is a line, as is easily
checked in local coordinates. The second of these defines a quadric in
$\mathbf{P}^{5}$ for degree reasons, and this intersects $\Pi$ in a plane
conic. Therefore, in the Grassmannian $\mathbf{G}(1,\Pi)\cong\mathbf{G}(1,3)$,
the set of lines meeting the four quadric surfaces in 8 distinct points is
also open, so $b=8$. ∎

######  Corollary 3.5.3.

The Plücker degree $\deg\overline{\mathcal{O}}_{5}=56$.

###### Proof.

Combine the previous result with $\deg\Sigma_{3}=4$ and $\deg\Sigma_{2,1}=5$.
∎

###  3.6 Orbit $\mathcal{O}_{6}$

This is the orbit of pencils which correspond to lines intersecting $X$ in two
distinct points.

######  Proposition 3.6.1.

The closure $\overline{\mathcal{O}}_{6}$ is the variety $\mathcal{S}X$ of
lines secant to $X$. It is birational to the Hilbert scheme
$\operatorname{Hilb}^{2}\mathbf{P}^{2*}$ of pairs of lines in the plane. In
particular, $\overline{\mathcal{O}}_{6}$ is irreducible of dimension 4.

###### Proof.

It follows almost by definition that
$\overline{\mathcal{O}}_{6}=\mathcal{S}X$, which is irreducible of dimension
$\dim\mathcal{S}X=2\dim X=4$ (see [14, Prop. 11.24]). The map
$\operatorname{Sym}^{2}X\smallsetminus\Delta_{X}\to\mathcal{S}X$ sending a
pair of points $\\{x,y\\}$ with $x\neq y$ to the line spanned by them is an
isomorphism onto $\mathcal{O}_{6}$ and hence gives us a birational map between
$\mathcal{S}X$ and the Hilbert scheme $\operatorname{Hilb}^{2}X$ of pairs on
points on $X$, which is simply $\operatorname{Hilb}^{2}\mathbf{P}V^{*}$. ∎

Now we have:

######  Proposition 3.6.2.

The class $[\overline{\mathcal{O}}_{6}]=3\sigma_{3,1}+6\sigma_{2,2}$.

###### Proof.

Write $[\overline{\mathcal{O}}_{6}]=a\sigma_{4}+b\sigma_{3,1}+c\sigma_{2,2}$.
To figure $a$ out, we take the product with $\sigma_{4}$, which amounts to
asking: given a general point $p\in\mathbf{P}^{5}$, how many lines through $p$
are secant to $X$? Well, the Veronese surface is famously defective and the
secant variety $S=S(X)$ is a proper subvariety of $\mathbf{P}^{5}$, so the
answer is simply $a=0$.

To figure $b$ out, we take the product with $\sigma_{3,1}$, and this amounts
to asking: given a general line $\ell\subset\mathbf{P}^{5}$ and a hyperplane
$H$ containing $\ell$, how many secant lines to $X$ intersect $\ell$ and lie
in $H$? By Bertini’s and Bezout’s theorems, for a general $H$ the intersection
$H\cap X$ is a rational normal curve of degree 4 and the intersection $H\cap
S$ is the secant variety of this rational normal curve. A general line
$\ell\subset H$ meets $H\cap S$ in $\deg S=3$ points. Generally, these points
will not lie on the tangential surface to $H\cap X$. Each of these points lies
on exactly one secant to the $H\cap X$, since otherwise we’d get 4 distinct
coplanar points on it, which is impossible. In all, it follows that there are
$b=3$ such secant lines intersecting $\ell$.

Finally, to figure $c$ out, we take the product with $\sigma_{2,2}$, and this
amounts to asking: given a general 3-plane $\Pi\subset\mathbf{P}^{5}$, how
many secant lines to $X$ lie in $\Pi$? As observed several times above, a
general 3-plane $\Pi$ intersects $X$ in 4 points in linear general position,
so in general there are $c=\binom{4}{2}=6$ such secant lines. ∎

######  Corollary 3.6.3.

The Plücker degree $\deg\overline{\mathcal{O}}_{6}=21$.

###### Proof.

Combine the previous result with $\deg\Sigma_{3,1}=3$ and
$\deg\Sigma_{2,2}=2$. ∎

###  3.7 Orbit $\mathcal{O}_{7}$

This is the orbit of pencils corresponding to lines $\ell\subset S$ not
meeting $X$ at all, and is an open dense subset of the component of the Fano
variety $F_{1}(S)$ other than $\mathcal{S}X$.

######  Proposition 3.7.1.

The variety $\overline{\mathcal{O}}_{7}$ is isomorphic to the Segre fourfold
$\mathbf{P}^{2}\times\mathbf{P}^{2*}$ and is in particular irreducible of
dimension 4.

###### Proof.

Consider the map $\mathbf{P}^{2}\times\mathbf{P}^{2*}\to\mathbf{G}(1,5)$
sending a pair $(p,L)$ to the pencil of lines containing the fixed line $L$
and lines rotating around $p$. This is an injective map which is an embedding
away from the locus $p\in L$. The image of this map is irreducible closed
subset of $\mathbf{G}(1,5)$ which contains $\mathcal{O}_{7}$ as an open dense
subset. ∎

######  Proposition 3.7.2.

The class $[\overline{\mathcal{O}}_{7}]=6\sigma_{3,1}+3\sigma_{2,2}$.

Again, we give two proofs of this result, one here and one in §4.

###### First Proof of Proposition 3.7.2..

Write $[\overline{\mathcal{O}}_{7}]=a\sigma_{4}+b\sigma_{3,1}+c\sigma_{2,2}$.
To figure $a$ out, we take the product with $\sigma_{4}$, which amounts to
asking: given a general point $p\in\mathbf{P}^{5}$, how many lines through $p$
belong to $\overline{\mathcal{O}}_{7}$? Since such lines are contained in $S$
and, as above, a general point $p$ doesn’t lie in $S$ at all, the answer is
$a=0$.

To figure $b$ out, we take the product with $\sigma_{3,1}$, which amounts to
asking: given a general line $\ell\subset\mathbf{P}^{5}$ and a hyperplane $H$
containing $\ell$, how many lines contained in the fourfold $S$ intersect
$\ell$ and lie in $H$, other than the $3$ secants found above? Well, as
before, the line $\ell$ meets $H\cap S$ in 3 points not lying on the tangent
surface to $H\cap X$. Through each of these points, as it turns out and as we
discuss below, there are exactly two lines contained in $S$ which are not
secant to $X$, and so this gives us a total of $b=6$ such lines.

Finally, to calculate $c$, we take the product with $\sigma_{2,2}$, and this
amounts to asking: given a general $3$-plane $\Pi\subset\mathbf{P}^{5}$, how
many lines in $\overline{\mathcal{O}}_{7}$ lie in $\Pi$? This corresponds to
lines in the Cayley cubic $\Pi\cap S$ other than the 6 lines through its 4
ordinary double points, and this is well-known to be $c=3$. ∎

The remaining piece of the link is the following claim:

######  Lemma 3.7.3.

Let $\Gamma\subset\mathbf{P}^{4}$ be the rational normal quartic. Given a
general point131313It suffices to take $p$ not lying on the tangent surface
$\tan(\Gamma)$. $p$ on the secant threefold $S_{\Gamma}:=S(\Gamma)$ to
$\Gamma$, there are exactly three lines through $p$ contained in $S_{\Gamma}$,
namely one line secant to $\Gamma$ and two other lines, from which $\Gamma$ is
projected $2:1$ onto a smooth conic in the complementary plane.

To put this into context, we recall the well-known fact that given any point
$p$ on a smooth cubic threefold $Y$, there are exactly 6 lines on $Y$ passing
through $p$. As this smooth cubic threefold specializes to the secant
threefold $S_{\Gamma}$ (see the family explained in [20, §2]), one can study
the specalization of this configuration of six lines. Segre showed in [21]
that the Fano variety $F_{1}(S_{\Gamma})$ has two components, namely the
variety $\mathcal{S}(\Gamma)$ of secant lines to $\Gamma$, and another
component of lines from which $\Gamma$ is projected $2:1$ onto a smooth conic
in the complementary plane; these two components intersect in the variety
$\mathcal{T}(\Gamma)$ of tangent lines to $\Gamma$. It turns out that the
first component has multiplicity 4 in the Fano scheme $F_{1}(S_{\Gamma})$ (a
proof is outlined in [20, §2]; we’ll calculate this multiplicity below in §4
as well), and so counted properly this tells us through a given general point
there should pass two lines of the second component. In stead of quoting this
bit of theory, we give an alternative self-contained proof here.

###### Proof of Lemma 3.7.3.

Think of $\mathbf{P}^{4}=\mathbf{P}H^{0}\mathcal{O}_{\mathbf{P}^{1}}(4)$ as
the space of binary quartic forms. The $j$-invariant of (the cross-ratio of)
the four roots of the polynomial (see footnote 1) gives a rational function
$j:\mathbf{P}^{4}\dashrightarrow\mathbf{P}^{1}$, whose fiber over a general
point is a sextic hypersurface. There are two exceptional fibers corresponding
to $j=0$ and $j=1728$, which are a triple quadric and a double cubic
respectively. In fact, from classical invariant theory, we know that if we
give $\mathbf{P}^{4}$ the coordinates $a_{0},\dots,a_{4}$ so that
$[a_{0},\dots,a_{4}]$ corresponds to the quartic polynomial
$a_{0}x^{4}+a_{1}x^{3}y+\cdots+a_{4}y^{4}$, then these quadric and cubic
surfaces are defined by the vanishing of the two invariants

$\displaystyle I$ $\displaystyle=12a_{0}a_{4}-3a_{1}a_{3}+a_{2}^{2}\text{ and
}$ $\displaystyle J$
$\displaystyle=72a_{0}a_{2}a_{4}-27a_{0}a_{3}^{2}-27a_{1}^{2}a_{4}+9a_{1}a_{2}a_{3}-2a_{2}^{3}=\frac{1}{4}\det\begin{bmatrix}12a_{0}&3a_{1}&2a_{2}\\\
3a_{1}&2a_{2}&3a_{3}\\\ 2a_{2}&3a_{3}&12a_{4}\end{bmatrix}$

respectively. The latter, which will be more relevant to us, is (up to scaling
the coordinates) the classical catalecticant or Hankel determinant of the
binary quartic.

The space of all fourth powers of linear forms is a rational normal quartic
$\Gamma\subset\mathbf{P}^{4}$. A quartic has $j$-invariant 1728 iff by a
$\operatorname{PGL}_{2}$ action its roots can be taken to $\\{\pm 1,\pm i\\}$
and this happens iff it is a linear combination of two fourth powers of linear
forms, from which it follows that the cubic three-fold $\mathbf{V}(J)$ is
exactly the secant threefold $S_{\Gamma}$ to $\Gamma$. Now let $L_{1},L_{2}$
be distinct linear forms on $\mathbf{P}^{1}$; then for any $\lambda\in
K^{\times}$, it is easy to see141414By a $\operatorname{PGL}_{2}$
transformation, we can assume that $L_{1}=x$ and $L_{2}=y$, say. from the
explicit equation $J$ defining $S_{\Gamma}$ that there are exactly three lines
in $S_{\Gamma}$ through the point $[L_{1}^{4}+\lambda L_{2}^{4}]\in
S_{\Gamma}$, namely the secant line itself and the two lines joining
$[L_{1}^{4}+\lambda L_{2}^{4}]$ to
$[L_{1}L_{2}(L_{1}^{2}\pm\sqrt{-\lambda}L_{2}^{2})]$. A general point on each
of these two latter lines lies on a unique secant to $\Gamma$, so projection
from such a line expresses $\Gamma$ as a two sheeted cover of a plane conic
for degree reasons. ∎

######  Corollary 3.7.4.

The Plücker degree $\deg\overline{\mathcal{O}}_{7}=24$.

###### Proof.

Combine the previous result with $\deg\Sigma_{3,1}=3$ and
$\deg\Sigma_{2,2}=2$. ∎

###  3.8 Orbit $\mathcal{O}_{8}$

This is the closed orbit of pencils corresponding to lines $\ell\subset S$
tangent to $X$.

######  Proposition 3.8.1.

The orbit $\mathcal{O}_{8}=\mathcal{T}X$ is the variety of lines tangent to
$X$. In particular, $\mathcal{O}_{8}$ is a $\mathbf{P}^{1}$ bundle over
$X\cong\mathbf{P}^{2}$ and so irreducible of dimension 3.

###### Proof.

Clear. ∎

######  Proposition 3.8.2.

The class $[\mathcal{O}_{8}]=6\sigma_{4,1}+6\sigma_{3,2}$.

###### Proof.

Write $[\mathcal{O}_{8}]=a\sigma_{4,1}+b\sigma_{3,2}$. To figure $a$ out, we
take the product with $\sigma_{3}$, which amounts to asking: given a general
line $\ell\subset\mathbf{P}^{5}$, how many lines intersecting $\ell$ are
tangent to the Veronese surface? Well, a general $\ell$ intersects the tangent
variety $\tan(X)=S(X)=S$ in $\deg S=3$ smooth points, and each such point lies
on exactly two tangents to the Veronese surface. Indeed, any point in
$S\smallsetminus X$ lies on a unique 2-plane contained in $S$ which intersects
$X$ in a plane conic. The existence of such a plane follows from the standard
argument that proves the defectiveness of the Veronese surface, see [14, Ch.
11]. The uniqueness of such a plane follows from the fact that points (or more
generally divisors) on rational normal curves are in linear general position,
see Lemma 10.14 in [22, §10.4]. Given this, the two tangents to $X$ through a
point $p\in S\smallsetminus X$ are exactly the two tangents to this mentioned
plane conic. It follows that there are $a=6$ such lines interesecting $\ell$
and tagent to $X$.

To figure $b$ out, we take the product with $\sigma_{2,1}$, which amounts to
asking: given a general 2-plane $\Pi\subset\mathbf{P}^{5}$ and a hyperplane
$H\subset\mathbf{P}^{5}$ containing $\Pi$, how many lines incident to $\Pi$
and contained $H$ are tangent to $X$? As above, the intersection $H\cap X$ is
a rational normal curve of degree 4. The tangent variety $\tan(H\cap X)$ to
such a curve is a surface of degree $2\cdot 4-2=6$, see [14, Ex. 19.11].
Therefore, a general 2-plane $\Pi\subset H$ intersects $\tan(H\cap X)$ in
exactly 6 points, each of which lies on a unique tangent to the rational
normal curve, again by the linear independence of divisors on rational normal
curves. Therefore, there are exactly $b=6$ such tangent lines in general. ∎

######  Corollary 3.8.3.

The Plücker degree $\deg\mathcal{O}_{8}=18$.

###### Proof.

Combine the previous result with $\deg\Sigma_{4,1}=1$ and
$\deg\Sigma_{3,2}=2$. ∎

##  4 Some Chern Class Computations

###  4.1 The Chow Ring Class of the Fano Scheme $F_{1}(S)$

The first proof of Proposition 3.7.2 suggests that we should calculate the
multiplicity of the irreducible components $\overline{\mathcal{O}}_{6}$ and
$\overline{\mathcal{O}}_{7}$ of the Fano scheme $F_{1}(S)$ of lines on $S$. We
claim:

######  Proposition 4.1.1.

The component $\overline{\mathcal{O}}_{6}$ of $F_{1}(S)$ has multiplicity 4
and the component $\overline{\mathcal{O}}_{7}$ has multiplicity 1.

###### Proof.

We know that every smooth cubic surface in $\mathbf{P}^{3}$ contains exactly
27 lines, and these can be seen explicitly by considering $S$ as the blowup of
the of the plane at a general configuration of 6 points151515The precise
condition is that the 6 points be in linear general position and that there is
no conic passing through them. See [16, §4.1] for more on this bit of
classical theory.; namely, these are the 6 exceptional divisors, the proper
transforms of the $\binom{6}{2}=15$ lines joining these points, and the proper
transforms of the six conics through all but one of the points. As this
general configuration of 6 points specializes to the 6 vertices of a complete
quadrilateral, it is easy to see how these lines specialize to the lines on
the Cayley cubic and with what multiplicity; namely, we get the 6 exceptional
divisors, each with multiplicity 4, and the three proper transforms of the
diagonals of the complete quadrilateral.161616See [16, §4.6] for more details.
These same multiplicities must therefore also hold for the Fano scheme
$F_{1}(S)$ of lines in $S$. ∎

Given this, we can write the class
$[F_{1}(S)]=4[\overline{\mathcal{O}}_{6}]+[\overline{\mathcal{O}}_{7}]\in
A\mathbf{G}(1,5)$. Therefore, if we can figure our $[F_{1}(S)]$, then we can
use this to give another calculation of $[\overline{\mathcal{O}}_{7}]$, since
we gave a self-contained calculation of $[\overline{\mathcal{O}}_{6}]$ in
Proposition 3.6.2 above.

######  Proposition 4.1.2.

The class of the Fano scheme $[F_{1}(S)]=18\sigma_{3,1}+27\sigma_{2,2}$.

###### Proof 1 of Proposition 4.1.2..

We use [22, Prop. 6.4]. Since $\dim F_{1}(S)=4$, its class in
$A\mathbf{G}(1,5)$ can be computed as the Chern class
$c_{4}(\operatorname{Sym}^{3}\mathcal{S}^{*})$, where
$\mathcal{S}\to\mathbf{G}(1,5)$ is the tautological bundle. We know that
$c(\mathcal{S}^{*})=1+\sigma_{1}+\sigma_{1,1}$. It follows exactly as in [22,
§6.2.1] from the splitting principle that

$c_{4}(\operatorname{Sym}^{3}\mathcal{S}^{*})=9\sigma_{1,1}(2\sigma_{1}^{2}+\sigma_{1,1})=18\sigma_{3,1}+27\sigma_{2,2}.$

∎

Now we have:

###### Proof 2 of Proposition 3.7.2.

We have

$[\overline{\mathcal{O}}_{7}]=[F_{1}(S)]-4[\overline{\mathcal{O}}_{6}]=18\sigma_{3,1}+27\sigma_{2,2}-4(3\sigma_{3,1}+6\sigma_{2,2})=6\sigma_{3,1}+3\sigma_{2,2}.$

∎

###  4.2 Chern Classes of Bundles of Relative Principal Parts

There is another construction that allows us to study the orbit closures
mentioned above and their classes, namely the bundle of relative principal
parts. This section will follow and use notation and terminology from [22,
§11.1]. Denote by $\Phi=\mathbf{P}\mathcal{S}$ the flag bundle that is the
projectivized universal subbundle over $\mathbf{G}(1,5)$, i.e. let

$\Phi:=\\{(\ell,p):p\in\ell\\}\subset\mathbf{G}(1,5)\times\mathbf{P}^{5},$

and let $\pi_{1}$ and $\pi_{2}$ denote its projections to $\mathbf{G}(1,5)$
and $\mathbf{P}^{5}$ respectively. Recall that the Chow ring $A(\Phi)$ of
$\Phi$ is given by

$A(\Phi)=A\mathbf{G}(1,5)[\zeta]/(\zeta^{2}-\sigma_{1}\zeta+\sigma_{1,1}),$

where $\zeta=c_{1}(\mathcal{O}_{\Phi}(1))=\pi_{2}^{*}\zeta_{\mathbf{P}^{5}}$
is the pullback of a hyperplane class (i.e. a generator) in
$A(\mathbf{P}^{5})$. Further, the pushforward map $\pi_{1,*}:A(\Phi)\to
A\mathbf{G}(1,5)$ is given by taking the coefficient of $\zeta$, i.e.

$\pi_{1,*}:A(\Phi)\to A\mathbf{G}(1,5),\quad\alpha\zeta+\beta\mapsto\alpha.$

For each $1\leq r\leq 4$, consider the bundle of relative principal parts

$\mathcal{E}^{r}:=\mathcal{P}^{r-1}_{\Phi/\mathbf{G}(1,5)}(\pi_{2}^{*}\mathcal{O}_{\mathbf{P}^{5}}(3))$

over $\Phi$ of rank $r$ whose fiber over a point $(\ell,p)$ is

$\mathcal{E}^{r}_{(\ell,p)}=H^{0}(\mathcal{O}_{\ell}(3)/\mathfrak{m}_{p}^{r}(3)).$

This admits a global section $\tau_{\Delta}\in H^{0}(\Phi,\mathcal{E}^{r})$
corresponding to the polynomial $\Delta$. The vanishing locus
$\mathbf{V}(\tau_{\Delta})\subset\Phi$ is then exactly the locus $\Phi_{r}$
defined in §2. Therefore, it follows that if
$\operatorname{codim}_{\Phi}\Phi_{r}=r$, then

$[\Phi_{r}]=c_{r}(\mathcal{E}^{r})\in A(\Phi)$

is the $r^{\text{th}}$ Chern class of $\mathcal{E}^{r}$. Finally, in the cases
where this holds and $\pi_{1}|_{\Phi_{r}}:\Phi_{r}\to\mathbf{G}(1,5)$ is
birational onto its image, then $\pi_{1,*}[\Phi_{r}]=[\pi_{1}(\Phi_{r})]$.
Therefore, computing the Chern class of $\mathcal{E}^{r}$ and taking its
pushforward should tell us something about the class of the orbit closures. To
compute the Chern class of $\mathcal{E}^{r}$, we use the filtration of
$\mathcal{E}^{r}$ with successive quotients

$\pi_{2}^{*}\mathcal{O}_{\mathbf{P}^{5}}(3),\pi_{2}^{*}\mathcal{O}_{\mathbf{P}^{5}}(3)\otimes\Omega_{\Phi/\mathbf{G}(1,5)},\cdots,\pi_{2}^{*}\mathcal{O}_{\mathbf{P}^{5}}(3)\otimes\operatorname{Sym}^{r-1}\Omega_{\Phi/\mathbf{G}(1,5)},$

where $\Omega_{\Phi/\mathbf{G}(1,5)}$ is the relative cotangent bundle of the
map $\Phi\to\mathbf{G}(1,5)$. Since

$c(\pi_{2}^{*}\mathcal{O}_{\mathbf{P}^{5}}(3))=1+3\zeta\text{ and
}c(\operatorname{Sym}^{m}\Omega_{\Phi/\mathbf{G}(1,5)})=1+m(\sigma_{1}-2\zeta),$

it follows that

$c(\mathcal{E}^{r})=\prod_{m=0}^{r-1}(1+(3-2m)\zeta+m\sigma_{1})$

so that

$c_{r}(\mathcal{E}^{r})=\prod_{m=0}^{r-1}((3-2m)\zeta+m\sigma_{1}).$

For $r=2,3,4$, these are easily seen to be

$\displaystyle c_{2}(\mathcal{E}^{2})$
$\displaystyle=6\sigma_{1}\zeta+3\sigma_{1,1},$ $\displaystyle
c_{3}(\mathcal{E}^{3})$ $\displaystyle=(6\sigma_{2}+9\sigma_{1,1})\zeta,\text{
and}$ $\displaystyle c_{4}(\mathcal{E}^{4})$
$\displaystyle=18\sigma_{3,1}+27\sigma_{2,2}$

From this we have:

###### Proof 3 of Proposition 3.2.2.

We showed in Proposition 3.2.1 that $\dim\Phi_{2}=7$, so that
$\operatorname{codim}_{\Phi}\Phi_{2}=\dim\Phi-\dim\Phi_{2}=9-7-2$. Further,
the projection $\pi_{1}:\Phi_{2}\to\overline{\mathcal{O}}_{2}$ is regular and
birational, and so it follows that

$[\overline{\mathcal{O}}_{2}]=[\pi_{1}(\Phi_{2})]=\pi_{1,*}[\Phi_{2}]=\pi_{1,*}(6\sigma_{1}\zeta+3\sigma_{1,1})=6\sigma_{1}.$

∎

###### Proof 2 of Proposition 3.4.2.

We showed in Proposition 3.4.2 that $\dim\Phi_{3}=6$, so that
$\operatorname{codim}_{\Phi}\Phi_{3}=3$. Further, the projection
$\pi_{1}:\Phi_{3}\to\overline{\mathcal{O}}_{4}$ is regular and birational, and
so it follows that

$[\overline{\mathcal{O}}_{4}]=[\pi_{1}(\Phi_{3})]=\pi_{1,*}[\Phi_{3}]=\pi_{1,*}\left((6\sigma_{2}+9\sigma_{1,1})\zeta\right)=6\sigma_{2}+9\sigma_{1,1}.$

∎

###### Proof 2 of Proposition 4.1.2.

The locus $\Phi_{4}=\pi_{2}^{-1}F(S)$ (as a scheme!), so that it follows that
$c_{4}(\mathcal{E}^{4})=[\Phi_{4}]=\pi_{2}^{*}[F(S)]$. Since $\pi_{2}^{*}$ is
injective, the result follows. ∎

##  5 Dealing with Characterstic $p>2$

In positive characteristic $p>2$, our results are still correct. Most of the
above arguments remain valid; the only thing that changes is that we have to
replace the use of Kleiman’s transversality theorem by an explicit calculation
of the generic transversality of the corresponding cycles. Since most of the
cycles we are interested in incidence varieties, varieties of secant lines,
varieties of tangent lines, or Fano varieties in the Grassmannian, we can use
the explicit description of the tangent spaces to such cycles (as found in say
[14, Lect. 16]) to check generic transversality. In this section, we check the
result for one orbit closure for which such description is not as easy to
write down (or locate a reference for), namely $\overline{\mathcal{O}}_{5}$.

Recall that $\mathcal{O}_{5}$ is the orbit of lines intersecting $S$ in a
unique point of $X$. In an affine neighborhood of a line
$\ell\in\mathcal{O}_{5}$, it is straightforward to write down defining
equations of the variety $\overline{\mathcal{O}}_{5}$; to help in this
process, you can take one point of such a line varying on $X$. Given this, it
is very easy to figure out the tangent space $T_{\ell}\mathcal{O}_{5}$ as a
subspace of $T_{\ell}\mathbf{G}(1,5)=\operatorname{Hom}(\ell,K^{6}/\ell)$. We
see that if $p\in\ell\cap X$ is the unique point of intersection of $\ell$
with $X$, then if $H_{\ell}\subset K^{6}$ is the hyperplane of conics passing
through the (underlying reduced subscheme, i.e. the point) of the base locus
$B_{\ell}$, then

$T_{\ell}\mathcal{O}_{5}=\\{\varphi\in\operatorname{Hom}(\ell,K^{6}/\ell):\varphi(p)\subseteq\mathbf{T}_{p}X+\ell\text{
and }\varphi(\ell)\subseteq H_{\ell}\\}.$

This amounts to saying that as we vary the line $\ell$ along the tangent
vector $\varphi$, to keep it incident to $X$ amounts to the condition
$\varphi(p)\subseteq\mathbf{T}_{p}X+\ell$, and to keep $\ell$ in the tangent
cone to $S$ at its point of intersection with $X$ amounts to the additional
condition $\varphi(\ell)\subseteq H_{\ell}$. From this, it follows that a
general complementary Schubert cycle is generically transverse to
$\mathcal{O}_{5}$, which is the needed generic transversality result.

##  6 Suggestions for Further Work

There are many natural ways to extend our results; we mention a few here.

  1. (a)

While $n=2,d=1$ is the last nontrivial case in which there are only finitely
many orbits of the $\operatorname{PGL}_{n+1}$ action on
$\mathbf{G}(d,n(n+3)/2)$, for higher $n$ and $d$ there are still certain
exceptional orbits, the classes in $A\mathbf{G}(d,n(n+3)/2)$ of the closures
of which can be calculated by methods similar to ours. For instance, one could
look at the Chow ring classes of the 14 exceptional orbits of nets of plane
conics ($n=d=2$) mentioned in [8].

  2. (b)

The same also applies to linear systems of cubics or higher degree
hypersurfaces. Even the case of plane cubics is already very interesting and
nontrivial, c.f. [22, §2.2] and the references there. In particular, see the
book [23] by Greuel, Lossen and Shustin for more on what is known in the
general case.

  3. (c)

Finally, one can try to find the analogous results to ours in characteristic
$p=2$, as Vainsencher does in [12] for the classical problem of the number of
smooth conics tangent to 5 general conics (the answer being $51$ as opposed to
$3264$ in characteristic $p\neq 2$).

## References

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# Deep Active Inference
for Autonomous Robot Navigation

Ozan Çatal, Samuel Wauthier, Tim Verbelen, Cedric De Boom, & Bart Dhoedt
IDLab, Department of Information Technology
Ghent University –imec
Ghent, Belgium
<EMAIL_ADDRESS>
###### Abstract

Active inference is a theory that underpins the way biological agent’s
perceive and act in the real world. At its core, active inference is based on
the principle that the brain is an approximate Bayesian inference engine,
building an internal generative model to drive agents towards minimal
surprise. Although this theory has shown interesting results with grounding in
cognitive neuroscience, its application remains limited to simulations with
small, predefined sensor and state spaces.

In this paper, we leverage recent advances in deep learning to build more
complex generative models that can work without a predefined states space.
State representations are learned end-to-end from real-world, high-dimensional
sensory data such as camera frames. We also show that these generative models
can be used to engage in active inference. To the best of our knowledge this
is the first application of deep active inference for a real-world robot
navigation task.

##  1 Introduction

Active inference and the free energy principle underpins the way our brain –
and natural agents in general – work. The core idea is that the brain
entertains a (generative) model of the world which allows it to learn cause
and effect and to predict future sensory observations. It does so by
constantly minimising its prediction error or “surprise”, either by updating
the generative model, or by inferring actions that will lead to less
surprising states. As such, the brain acts as an approximate Bayesian
inference engine, constantly striving for homeostasis.

There is ample evidence (Friston, 2012; Friston et al., 2013a; 2014) that
different regions of the brain actively engage in variational free energy
minimisation. Theoretical grounds indicate that even the simplest of life
forms act in a free energy minimising way (Friston, 2013).

Although there is a large body of work on active inference for artificial
agents (Friston et al., 2006; 2009; 2017; 2013b; Cullen et al., 2018),
experiments are typically done in a simulated environment with predefined and
simple state and sensor spaces. Recently, research has been done on using deep
neural networks as an implementation of the active inference generative model,
resulting in the umbrella term “deep active inference”. However, so far all of
these approaches were only tested on fairly simple, simulated environments
(Ueltzhöffer, 2018; Millidge, 2019; Çatal et al., 2019). In this paper, we
apply deep active inference on a robot navigation task, with high-dimensional
camera observations and deploy it on a mobile robot platform. To the best of
our knowledge, this is the first time that active inference is applied on a
real-world robot navigation task.

In the remainder of this paper we will first introduce the active inference
theory in Section 2. Next, we show how we implement active inference using
deep neural networks in Section 3, and discuss initial experiments in Section
4.

##  2 Active Inference

Active inference is a process theory of the brain that utilises the concept of
free energy (Friston, 2013) to describe the behaviour of various agents. It
stipulates that all agents act in order to minimise their own uncertainty of
the world. This uncertainty is expressed as Bayesian Surprise, or
alternatively the variational free energy. In this context this is
characterised by the difference between what an agent imagines about the world
and what it has perceived about the world (Friston, 2010). More concretely,
the agent builds a generative model
$P(\tilde{{\bm{o}}},\tilde{{\bm{s}}},\tilde{{\bm{a}}})$, linking together the
agents internal belief states ${\bm{s}}$ with the perceived actions ${\bm{a}}$
and observations ${\bm{o}}$ in the form of a joint distribution. We use a
tilde to denote a sequence of variables through time. This generative model
can be factorised as in Equation 1.

$P(\tilde{{\bm{o}}},\tilde{{\bm{s}}},\tilde{{\bm{a}}})=P(\tilde{{\bm{a}}})P({\bm{s}}_{0})\prod_{t=1}^{T}P({\bm{o}}_{t}|{\bm{s}}_{t})P({\bm{s}}_{t}|{\bm{s}}_{t-1},{\bm{a}}_{t-1})$
(1)

The free energy or Bayesian surprise is then defined as:

$\begin{split}F&=\mathbb{E}_{Q}[\log Q(\tilde{{\bm{s}}})-\log
P(\tilde{{\bm{o}}},\tilde{{\bm{s}}},\tilde{{\bm{a}}})]\\\
&=D_{\mathrm{KL}}(Q(\tilde{{\bm{s}}})\|P(\tilde{{\bm{s}}},\tilde{{\bm{a}}}|\tilde{{\bm{o}}}))-\log
P(\tilde{{\bm{o}}})\\\
&=D_{\mathrm{KL}}(Q(\tilde{{\bm{s}}})\|P(\tilde{{\bm{s}}},\tilde{{\bm{a}}}))-\mathbb{E}_{Q}[\log
P(\tilde{{\bm{o}}}|\tilde{{\bm{s}}})]\end{split}$ (2)

Here, $Q(\tilde{{\bm{s}}})$ is an approximate posterior distribution. The
second equality shows that free energy is equivalent to the (negative)
evidence lower bound (ELBO) (Kingma & Welling, 2013; Rezende et al., 2014).
The final equation frames the problem of free energy minimisation as
explaining the world from the agents beliefs whilst minimising the complexity
of accurate explanations (Friston et al., 2016).

Crucially, in active inference agents will act according to the belief that
they will keep minimising surprise in the future. This means agents will infer
policies that yield minimal expected free energy in the future, with a policy
$\pi$ being the sequence of future actions ${\bm{a}}_{t:t+H}$ starting at
current time step $t$ with a time horizon $H$. This principle is formalised in
Equation 3 with $\sigma$ being the softmax function with precision parameter
$\gamma$.

$\begin{split}P(\pi)&=\sigma(-\gamma G(\pi))\\\
G(\pi)&=\sum_{\tau=t}^{t+H}G(\pi,\tau)\end{split}$ (3)

Expanding the expected free energy functional $G(\pi,\tau)$ we get Equation 4.
Using the factorisation of the generative model from Equation 1 we approximate
$Q({\bm{o}}_{\tau},{\bm{s}}_{\tau}|\pi)\approx
P({\bm{o}}_{\tau}|{\bm{s}}_{\tau})Q({\bm{s}}_{\tau}|\pi)$.

$\begin{split}G(\pi,\tau)&=\mathbb{E}_{Q({\bm{o}}_{\tau},{\bm{s}}_{\tau}|\pi)}[\log
Q({\bm{s}}_{\tau}|\pi)-\log P({\bm{o}}_{\tau},{\bm{s}}_{\tau}|\pi)]\\\
&=\mathbb{E}_{Q({\bm{o}}_{\tau},{\bm{s}}_{\tau}|\pi)}[\log
Q({\bm{s}}_{\tau}|\pi)-\log P({\bm{o}}_{\tau}|{\bm{s}}_{\tau},\pi)-\log
P({\bm{s}}_{\tau}|\pi)]\\\
&=D_{\mathrm{KL}}(Q({\bm{s}}_{\tau}|\pi)\|P({\bm{s}}_{\tau}))+\mathbb{E}_{Q({\bm{s}}_{\tau})}[H(P({\bm{o}}_{\tau}|{\bm{s}}_{\tau}))]\end{split}$
(4)

Note that, in the final equality, we substitute $P({\bm{s}}_{\tau}|\pi)$ by
$P({\bm{s}}_{\tau})$, a global prior distribution on the so-called “preferred”
states of the agent. This reflects the fact that the agent has prior
expectations about the states it will reach. Hence, minimising expected free
energy entails both realising preferences, while minimising the ambiguity of
the visited states.

##  3 Deep active inference

Figure 1: The various components of the agent rolled out trough time. We
minimise the variational free energy by minimising both the negative log
likelihood of observations and the KL divergence between the state transition
model and the observation model. The inferred hidden state is characterised as
a multivariate Gaussian distribution.

In current treatments of active inference the state spaces are typically
completely fixed upfront (Friston et al., 2009; Millidge, 2019) or partially
(Ueltzhöffer, 2018). However, this does not scale well for more complex tasks
as it is often difficult to design meaningful state spaces for such problems.
Therefore we allow for the agent to learn by itself what the exact
parameterisation of its belief space should be. We enable this by using deep
neural networks to generate the various necessary probability distributions
for our agent.

We approximate the variational posterior distribution for a _single_ timestep
$Q({\bm{s}}_{t}|{\bm{s}}_{t-1},{\bm{a}}_{t-1},{\bm{o}}_{t})$ with a network
$q_{\phi}({\bm{s}}_{t}|{\bm{s}}_{t-1},{\bm{a}}_{t-1},{\bm{o}}_{t})$. Similarly
we approximate the likelihood model $P({\bm{o}}_{t}|{\bm{s}}_{t})$ with the
network $p_{\xi}({\bm{o}}_{t}|{\bm{s}}_{t})$ and the prior
$P({\bm{s}}_{t}|{\bm{s}}_{t-1},{\bm{a}}_{t-1})$ with the network
$p_{\theta}({\bm{s}}_{t}|{\bm{s}}_{t-1},{\bm{a}}_{t-1})$. Each of the networks
output a multivariate normal distribution with a diagonal covariance matrix
using the reparameterisation trick (Kingma & Welling, 2013). These neural
networks cooperate in a way similar to a VAE, where the fixed standard normal
prior is replaced with the learnable prior $p_{\theta}$, the decoder by
$p_{\xi}$ and finally the encoder by $q_{\phi}$, as visualised in Figure 1.

These networks are trained end-to-end using the free energy formula from the
previous section as an objective.

$\forall t:\underset{\phi,\theta,\xi}{\text{minimise}}:-\log
p_{\xi}({\bm{o}}_{t}|{\bm{s}}_{t})+D_{\mathrm{KL}}(q_{\phi}({\bm{s}}_{t}|{\bm{s}}_{t-1},{\bm{a}}_{t-1},{\bm{o}}_{t})\|p_{\theta}({\bm{s}}_{t}|{\bm{s}}_{t-1},{\bm{a}}_{t-1}))$
(5)

As in a conventional VAE (Kingma & Welling, 2013) the negative log likelihood
(NLL) term in the objective punishes reconstruction error forcing the model to
learn relevant information on the belief state to be captured in the posterior
output, while the KL term pulls the prior output towards the posterior output,
forcing the prior and posterior to agree on the content of the belief state in
a way that still allows the likelihood model to reconstruct the current
observation.

We can now use the learned models to engage in active inference, and infer
which action the agent has to take next. This is done by generating imagined
trajectories for different policies using $p_{\theta}$ and $p_{\xi}$,
calculating the expected free energy $G$ and selecting the action of the
policy that yields the lowest $G$. These policies to evaluate can be
predefined, or generated through random shooting, using cross-entropy method
(Boer et al., 2005) or by building a search tree.

##  4 Experiments

We validate our deep active inference approach on a real world robotics
navigation task. First, we collect a dataset consisting of two hours worth of
real world action-observation sequences by driving a Kuka Youbot base platform
up and down the aisles of a warehouse lab. Camera observations are recorded
with a front mounted Intel Realsense RGB-D camera, without taking into account
the depth information. The x, y and angular velocities are recorded as actions
at a recording frequency of 10Hz. The models are trained on a subsampled
version of the data resulting in a train set with data points every 200ms.

Next, we instantiate neural networks $q_{\phi}$ and $p_{\xi}$ as a
convolutional encoder and decoder network, and $p_{\theta}$ using an LSTM.
These are trained with Adam optimizer using the objective function from
Equation 5 for 1M iterations. We use a minibatch size of 128 and a sequence
length of 10 timesteps. A detailed overview of all hyperparameters is given in
appendix.

We utilise the same approach as in Çatal et al. (2020) for our imaginary
trajectories and planning. The agent has access to three base policies to pick
from: drive straight, turn left and turn right. Actions from these policies
are propagated to the learned models at different time horizons $H=10,25$ or
$55$. For each resulting imaginary trajectory, the expected free energy $G$ is
calculated. Finally the trajectory with lowest $G$ is picked, and the first
action of the chosen policy is executed, after which the imaginary planning
restarts. The robot’s preferences are given by demonstration, using the state
distribution of the robot while driving in the middle of the aisle. This
should encourage the robot to navigate in the aisles.

At each trial the robot is placed at a random starting position and random
orientation and tasked to navigate to the preferred position. Figure 2
presents a single experiment as an illustrative example. Figure 2(a) shows the
reconstructed preferred observation from the given preferred state, while
Figure 2(b) shows the trial’s start state from an actual observation. Figure
2(c) shows the imagined results of either following the policy “always turn
right”, “always go straight” or “always turn left”. Figure 2(d) is the result
of utilising the planning method explained above. Additional examples can be
found in the supplementary material.

The robot indeed turns and keeps driving in the middle of the aisle, until it
reaches the end and then turns around 111A movie demonstrating the results is
available at https://tinyurl.com/smvyk53. When one perturbs the robot by
pushing it, it will again recover and continue to the middle of the aisle.

(a) Preferred state.

(b) Start state

(c) Imaginary future trajectories for different policies, i.e. going straight
ahead (top), turning right (middle), turning left (bottom).

(d) Actually followed trajectory.

Figure 2: Experimental results: Figure (a) shows the target observation in
imagined (reconstructed) space. (b) The start observation of the trial. Figure
(c) shows different imaginary planning results, whilst (d) shows the actually
followed trajectory.

##  5 Conclusion

In this paper we present how we can implement a generative model for active
inference using deep neural networks. We show that we are able to successfully
execute a simple navigation task on a real world robot with our approach. As
future work we want to allow the robot to continuously learn from past
autonomous behaviour, effectively “filling the gaps” in its generative model.
Also how to define the “preferred state” distributions and which policies to
evaluate remains an open research challenge for more complex tasks and
environments.

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##  Appendix A Neural architecture

| Layer | Neurons/Filters | activation function
---|---|---|---
Posterior | Convolutional | 8 | Leaky ReLU
Convolutional | 16 | Leaky ReLU
Convolutional | 32 | Leaky ReLU
Convolutional | 64 | Leaky ReLU
Convolutional | 128 | Leaky ReLU
Concat | N.A. | N.A.
Linear | 2 x 128 states | Softplus
Likelihood | Linear | 128 x 8 x 8 | Leaky ReLU
Convolutional | 128 | Leaky ReLU
Convolutional | 64 | Leaky ReLU
Convolutional | 32 | Leaky ReLU
Convolutional | 16 | Leaky ReLU
Convolutional | 8 | LeakyReLU
Prior | LSTM cell | 400 | Leaky ReLU
Linear | 2 x128 states | Softplus
Table 1: Neural network architectures. All convolutional layers have a 3x3
kernel. The convolutional layers in the Likelihood model have a stride and
padding of 1 to ensure that they preserve the input shape. Upsampling is done
by nearest neighbour interpolation. The concat step concatenates the processed
image pipeline with the vector inputs ${\bm{a}}$ and ${\bm{s}}$.

##  Appendix B Hyperparameters

| Parameter | Value
---|---|---
Learning | learning rate | 0.0001
batch size | 128
train iterations | 1M
sequence length | 10
Planning | $\gamma$ | 100
D (Çatal et al., 2020) | 1
K (Çatal et al., 2020) | 10, 25, 55
N (Çatal et al., 2020) | 5
$\rho$ (Çatal et al., 2020) | 0.001
Table 2: Overview of the model hyperparemeters.

##  Appendix C Detailed Planning example

A movie demonstrating the results is available at https://tinyurl.com/smvyk53.

Figure 3: Trial preferred state Figure 4: Short term planning

Figure 5: Middle long term planning

Figure 6: Long term planning