# Bridgeland stability conditions and skew lines on $\mathbb{P}^{3}$

Sammy Alaoui Soulimani and Martin G. Gulbrandsen Sammy Alaoui Soulimani Email:
<EMAIL_ADDRESS>G. Gulbrandsen Email<EMAIL_ADDRESS>University of Stavanger, Department of Mathematics and Physics, 4036
Stavanger, Norway

###### Abstract

Inspired by Schmidt’s work on twisted cubics [20], we study wall crossings in
Bridgeland stability, starting with the Hilbert scheme
$\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$ parametrizing pairs of skew lines and
plane conics union a point. We find two walls. Each wall crossing corresponds
to a contraction of a divisor in the moduli space and the contracted space
remains smooth. Building on work by Chen–Coskun–Nollet [8] we moreover prove
that the contractions are $K$-negative extremal in the sense of Mori theory
and so the moduli spaces are projective.

##  1 Introduction

After the introduction of Bridgeland’s manifold of stability conditions on a
triangulated category [6], several applications to the study of the birational
geometry of moduli spaces have appeared: the moduli space is viewed as
parametrizing stable objects in the derived category of some underlying
variety $X$, and the question is how the moduli space changes as the stability
condition varies. This is the topic of wall crossing in the stability
manifold. We refer to [15] for an overview and in particular for examples of
the success of this viewpoint in cases where $X$ is a surface. For threefolds
and notably in the case $X=\mathbb{P}^{3}$, important progress was made by
Schmidt [20], allowing among other things a study of wall crossings for the
Hilbert scheme of twisted cubics (see also Xia [22] for further work on this
case; additional examples in the same spirit have been investigated by
Gallardo–Lozano Huerta–Schmidt [11] and Rezaee [19]). The case considered in
the present text is that of pairs of skew lines in $\mathbb{P}^{3}$ and their
deformations. This is analogous to twisted cubics in the sense that a twisted
cubic degenerates to a plane nodal curve with an embedded point much as a pair
of skew lines degenerates to a pair of lines in a plane together with an
embedded point.

More precisely we study wall crossing for the Hilbert scheme
$\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$ of subschemes $Y\subset\mathbb{P}^{3}$
with Hilbert polynomial $2m+2$. It has two smooth components $\mathcal{C}$ and
$\mathcal{S}$: a general point in $\mathcal{C}$ is a conic-union-a-point
$Y=C\cup\\{P\\}$ and a general point in $\mathcal{S}$ is a pair of skew lines
$Y=L_{1}\cup L_{2}$. Note that when a line pair is deformed until the two
lines meet, the result is a pair of intersecting lines with an embedded point
at the intersection, and this can also be viewed as a degenerate case of a
conic union a point.

For an appropriately chosen Bridgeland stability condition on the bounded
derived category of coherent sheaves $D^{b}(\mathbb{P}^{3})$, the ideal
sheaves $\mathcal{I}_{Y}$ can be viewed as the stable objects with fixed Chern
character, say $v=\mathrm{ch}(\mathcal{I}_{Y})$. When deforming the stability
condition, we identify two walls, separating three chambers. Getting slightly
ahead of ourselves, the situation is illustrated in Figure 2 in Section 3.2:
$\alpha$ and $\beta$ are parameters for the stability conditions considered
and we restrict ourselves to the region to the immediate left in the picture
of the hyperbola $\beta^{2}-\alpha^{2}=4$ (the role of this boundary curve is
explained in Section 2.4). In this region we have the two walls $W_{1}$ and
$W_{2}$ separating three chambers, labeled by Roman numerals as in the figure.
Let $\mathcal{M}^{\mathrm{I}}$, $\mathcal{M}^{\mathrm{II}}$ and
$\mathcal{M}^{\mathrm{III}}$ be the moduli spaces of Bridgeland stable objects
with Chern character $v$ in each chamber, considered as algebraic spaces (for
existence see Piyaratne–Toda [18, Corollary 4.23]).

Our first main result contains the set-theoretical description of these moduli
spaces:

######  Theorem 1.1.

  1. (I)

$\mathcal{M}^{\mathrm{I}}$ is $\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$ with its
two components $\mathcal{S}$ and $\mathcal{C}$ described above.

  2. (II)

$\mathcal{M}^{\mathrm{II}}$ consists of:

    1. (i)

Ideal sheaves $\mathcal{I}_{Y}$ for $Y\in\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$
not contained in a plane.

    2. (ii)

Non-split extensions $\mathcal{F}_{P,V}$ in

$0\to\mathcal{I}_{P/V}(-2)\to\mathcal{F}_{P,V}\to\mathcal{O}_{\mathbb{P}^{3}}(-1)\to
0$ (1.1)

for $V\subset\mathbb{P}^{3}$ a plane and $P\in V$. Moreover,
$\mathcal{F}_{P,V}$ is uniquely determined up to isomorphism by the pair
$(P,V)$.

  3. (III)

$\mathcal{M}^{\mathrm{III}}$ consists of:

    1. (i)

Ideal sheaves $\mathcal{I}_{Y}$ for $Y\in\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$
a pair of disjoint lines or a pure double line.

    2. (ii)

Non-split extensions $\mathcal{G}_{P,V}$ in

$0\to\mathcal{O}_{V}(-2)\to\mathcal{G}_{P,V}\to\mathcal{I}_{P}(-1)\to 0$ (1.2)

with $P$ and $V$ as above. Moreover, $\mathcal{G}_{P,V}$ is uniquely
determined by the pair $(P,V)$.

To locate walls and classify stable objects we employ the method due to
Schmidt [20], which involves “lifting” walls from an intermediate notion of
tilt stability. Schmidt considers as an application the Hilbert scheme
$\mathrm{Hilb}^{3m+1}(\mathbb{P}^{3})$: it parametrizes twisted cubics and
plane cubics union a point. This was our starting point and we can apply many
of Schmidt’s results directly, although modified or new arguments are needed
as well. The end result is closely analogous in the two cases, with two wall
crossings of the same nature. In the twisted cubic situation, however, Schmidt
also finds an additional “final wall crossing” where all objects are
destabilized. This has no direct analogy in our case.

Next we describe the moduli spaces geometrically, guided by the set theoretic
classification of objects above; this leads to contractions of the two smooth
components $\mathcal{C}$ and $\mathcal{S}$ of
$\mathcal{M}^{\mathrm{I}}=\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$. First
introduce notation for the loci that are destabilized by the two wall
crossings according to the above classification:

######  Notation 1.2.

  * (a)

Let $E\subset\mathcal{C}$ be the divisor consisting of all planar
$Y\in\mathcal{C}$.

  * (b)

Let $F\subset\mathcal{S}$ be the divisor consisting of all $Y\in\mathcal{S}$
having an embedded point.

Thus the locus (II)(i) is $(\mathcal{C}\setminus E)\cup\mathcal{S}$ and the
locus (III)(i) is $\mathcal{S}\setminus F$. On the other hand both loci
(II)(ii) and (III)(ii) are parametrized by the incidence variety
$I\subset\mathbb{P}^{3}\times\check{\mathbb{P}}^{3}$ consisting of pairs
$(P,V)$ of points $P\in\mathbb{P}^{3}$ inside a plane
$V\in\check{\mathbb{P}}^{3}$. The process of replacing $E$ and $F$ by $I$ can
be realized as contractions of algebraic spaces: $E$ and $F$ may be viewed as
projective bundles over $I$, and in Section 4 we apply Artin’s contractibility
criterion to obtain smooth algebraic spaces $\mathcal{C}^{\prime}$ and
$\mathcal{S}^{\prime}$ each containing the incidence variety $I$ as a closed
subspace, and birational morphisms

$\displaystyle\phi$ $\displaystyle\colon\mathcal{C}\to\mathcal{C}^{\prime}$
$\displaystyle\psi$ $\displaystyle\colon\mathcal{S}\to\mathcal{S}^{\prime}$

which are isomorphisms outside of $E$, respectively $F$, and restrict to the
natural maps $E\to I$, respectively $F\to I$. Moreover $E\subset\mathcal{C}$
is disjoint from $\mathcal{S}$, so the union
$\mathcal{C}^{\prime}\cup\mathcal{S}$ makes sense as the gluing together of
$(\mathcal{C}\setminus E)\cup\mathcal{S}$ and $\mathcal{C}^{\prime}$. We can
then state our second main result:

######  Theorem 1.3.

  * (a)

$\mathcal{M}^{\mathrm{II}}$ is isomorphic to
$\mathcal{C}^{\prime}\cup\mathcal{S}$.

  * (b)

$\mathcal{M}^{\mathrm{III}}$ is isomorphic to $\mathcal{S}^{\prime}$.

To prove the theorem it suffices to treat the contracted spaces as algebraic
spaces. However, they turn out to be projective varieties: the contractions
are in fact $K$-negative extremal contractions in the sense of Mori theory.
The case of $\mathcal{S}\to\mathcal{S}^{\prime}$ can be found in previous work
by Chen–Coskun–Nollet [8] and in fact it turns out that $\mathcal{S}^{\prime}$
is a Grassmannian; see Section 4.2. Inspired by this work, we exhibit in
Section 5 the map $\mathcal{C}\to\mathcal{C}^{\prime}$ as a $K$-negative
extremal contraction. This may be contrasted with Schmidt’s approach in the
twisted cubic situation [20], where projectivity of the moduli spaces is
proved by viewing them as moduli of quiver representations.

In Section 2, we list the background results that we need, in particular, we
briefly recall the construction of stability conditions on threefolds, along
with the notion of tilt-stability. In Section 3 we apply Schmidt’s machinery
to prove Theorem 1.1. In Section 4 we study universal families and prove
Theorem 1.3. Finally, in Section 5 we work out the Mori cone of $\mathcal{C}$.

We work over $\mathbb{C}$. Throughout and in particular in Section 4,
intersections and unions of subschemes are defined by the sum and intersection
of ideals, respectively, and inclusions and equalities between subschemes are
meant in the scheme theoretic sense. The _relative ideal_ of an inclusion
$Z\subset Y$ of two closed subschemes of some ambient scheme is the ideal
$\mathcal{I}_{Z/Y}\subset\mathcal{O}_{Y}$ defining $Z$ as a subscheme of $Y$.

## Acknowledgments

This work is supported by the Research Council of Norway under grant no.
230986, and is part of the Ph.D. thesis [1] defended by the first author at
the university of Stavanger on the 29th of March 2023\. The second author was
also supported by the Swedish Research Council under grant no. 2016-06596
while the author was in residence at Institut Mittag–Leffler in Djursholm,
Sweden during the fall of 2021.

##  2 Preliminaries

After detailing the two components of $\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$,
we collect notions and results from the literature surrounding Bridgeland
stability and wall crossings for smooth projective threefolds. There are no
original results in this section.

###  2.1 The Hilbert scheme and its two components

It is known that $\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$ has two smooth
components $\mathcal{C}$ and $\mathcal{S}$, whose general points are conics
union a point and pairs of skew lines, respectively. A quick parameter count
yields $\dim\mathcal{C}=11$ and $\dim\mathcal{S}=8$. We refer to Lee [13] for
an overview, to Chen–Nollet [9] for the smoothness of $\mathcal{C}$ and to
Chen–Coskun–Nollet [8] for the smoothness of $\mathcal{S}$. In fact, the
referenced works show that $\mathcal{C}$ is the blowup

$\mathcal{C}\to\mathbb{P}^{3}\times\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})$

along the universal conic
$\mathcal{Z}\subset\mathbb{P}^{3}\times\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})$
and $\mathcal{S}$ is the blowup

$\mathcal{S}\to\mathrm{Sym}^{2}(G(2,4))$

along the diagonal in the symmetric square of the Grassmannian $G(2,4)$ of
lines in $\mathbb{P}^{3}$. In other words, it is the Hilbert scheme
$\mathrm{Hilb}^{2}(G(2,4))$ of finite subschemes in $G(2,4)$ of length two.

We pause to make some preparations regarding embedded points: by a curve
$C\subset\mathbb{P}^{3}$ _with an embedded point_ at $P\in C$ we mean a
subscheme $Y\subset\mathbb{P}^{3}$ such that $C\subset Y$ and the relative
ideal $\mathcal{I}_{C/Y}$ is isomorphic to $k(P)$. This makes sense even when
we allow $C$ to be singular or nonreduced. Embedded points are in bijection
with normal directions, i.e. lines in the normal space to
$C\subset\mathbb{P}^{3}$ at $P$: explicitly in our situation, suppose that
$P=(0,0,0)$ in local affine coordinates $x,y,z$ and $C$ is a conic defined by
the ideal $I_{C}=(q(x,y),z)$, where $q$ is a quadric vanishing at the origin.
A normal vector to $C$ may be viewed as a $k$-linear map

$\phi\colon I_{C}/\mathfrak{m}_{P}I_{C}\to k$

given by say $a=\phi(q)$ and $b=\phi(z)$. Thus $(a:b)\in\mathbb{P}^{1}$
parametrizes normal directions and the corresponding scheme $Y$ with an
embedded point at $P$ is defined by the kernel of the induced map $I_{C}\to
k$, which is

$I_{Y}=\mathfrak{m}_{P}I_{C}+(bq-az).$

######  Example 2.1.

With notation as above, consider the degenerate conic defined by $q(x,y)=xy$.
When $(a:b)=(1:0)$ we obtain the planar scheme $Y$ in the $xy$-plane given by

$I_{Y}=(x^{2}y,xy^{2},z)=(xy,z)\cap(z,x^{2},y^{2})$

where the final form exhibits $Y$ as the scheme theoretic union of a pair of
lines and a thickening of the origin in the $xy$-plane. At the other extreme
$(a:b)=(0:1)$ we obtain

$I_{Y}=(zy,zx,yz,z^{2})=(xy,z)\cap(x,y,z)^{2}$

which we label _spatial_ : the subscheme $Y\subset\mathbb{P}^{3}$ is not
contained in any nonsingular surface since it contains a full first order
infinitesimal neighbourhood around $P$. It is easy to check that for the
remaining values of $(a:b)$ the corresponding scheme $Y$ is neither planar nor
spatial. Analogous observations hold for the double line $q(x,y)=y^{2}$.

With these preparations, we next list all elements
$Y\in\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$, including degenerate cases. We
follow Lee [13, Section 3.5], to which we refer for further details and proof
that the following list is exhaustive.

The component $\mathcal{C}$ parametrizes subschemes $Y$ of the following form:
let $C$ be a conic in a plane $V\subset\mathbb{P}^{3}$, possibly a union of
two lines or a planar double line. Then $Y$ is either the disjoint union of
$C$ and a point $P\in\mathbb{P}^{3}$, or $C$ with an embedded point at $P\in
C$. If $C$ is nonsingular at $P$, embedded points correspond to normal
directions, parametrized by a $\mathbb{P}^{1}$. Since even degenerate conics
are complete intersections, also embedded point structures at a singular or
nonreduced point $P$ form a $\mathbb{P}^{1}$ as in Example 2.1, and among
these exactly one is planar ($Y$ contained in a plane) and exactly one is
spatial ($Y$ contains the first order infinitesimal neighbourhood of $P$ in
$\mathbb{P}^{3}$).

The component $\mathcal{S}$ parametrizes pairs $Y=L_{1}\cup L_{2}$ of skew
lines, together with its degenerations. These are of the following three
types: (1) a pair of incident lines $L_{1}\cup L_{2}$ with a spatial embedded
point at the intersection point, (2) a planar double line with a spatial
embedded point, or (3) a double line in a quadric surface, i.e. there is a
line $L$ in a nonsingular quadric surface $Q$ such that $Y$ is the effective
divisor $2L\subset Q$, but viewed as a subscheme of $\mathbb{P}^{3}$. We label
this case the _pure double line_ , where purity refers to the lack of embedded
components.

Clearly, then, $\mathcal{C}\cap\mathcal{S}$ consists of the incident or planar
double lines with a spatial embedded point.

###  2.2 Stability conditions and walls

Let $X$ be a smooth projective threefold over $\mathbb{C}$ and fix a finite
rank lattice $\Lambda$ equipped with a homomorphism $K(X)\to\Lambda$ from the
Grothendieck group of coherent sheaves modulo short exact sequences. On
$\mathbb{P}^{3}$ we will take
$\Lambda=\mathbb{Z}\oplus\mathbb{Z}\oplus\tfrac{1}{2}\mathbb{Z}\oplus\tfrac{1}{6}\mathbb{Z}$
equipped with the Chern character map $\mathrm{ch}\colon
K(\mathbb{P}^{3})\to\Lambda$.

Recall [6, 5, 4] that a Bridgeland stability condition
$\sigma=(\mathcal{A},Z)$ on $X$ (with respect to $\Lambda$) consists of

  * (i)

an abelian subcategory $\mathcal{A}\subset D^{b}(X)$, which is the heart of a
bounded $t$-structure, and

  * (ii)

a stability function $Z$, which is a group homomorphism

$Z\colon\Lambda\to\mathbb{C}$

whose value on any nonzero object $\mathcal{E}\in\mathcal{A}$ is in the upper
half plane

$\mathbb{H}=\big{\\{}z\in\mathbb{C}\;\vline\;\Im z\geq
0\big{\\}}\setminus\mathbb{R}_{\geq 0}.$

This is subject to a list of axioms which we will not give (see [4, Section
8]). We may then partially order the nonzero objects in $\mathcal{A}$ by their
_slope_

$\lambda_{\sigma}=-\Re(Z)/\Im(Z)\in\mathbb{R}\cup\\{+\infty\\}.$

This yields a notion of $\sigma$-stability and $\sigma$-semistability for
objects in $\mathcal{A}$ in the usual way by comparing the slope of an object
with that of its sub- or quotient objects. These notions extend to $D^{b}(X)$
by shifting in the sense of the $t$-structure.

Bridgeland’s result [6, Theorem 1.2] gives the set
$\mathrm{Stab}_{\Lambda}(X)$ of stability conditions the structure of a
complex manifold, and for a given $u\in\Lambda$, it admits a wall and chamber
structure: if an object $\mathcal{E}$ goes from being stable to being unstable
as the stability condition $\sigma$ varies, $\sigma$ has to pass through a
stability condition for which $\mathcal{E}$ is strictly semistable. This
observation leads to the definition of a wall:

######  Definition 2.2.

Fix $u\in\Lambda$.

  1. i.

Numerical walls: A numerical wall is a nontrivial proper solution set

$W_{v,w}=\\{\sigma\in\mathrm{Stab}_{\Lambda}(X)\;\vline\;\lambda_{\sigma}(v)=\lambda_{\sigma}(w)\\}$

where $u=v+w$ in $\Lambda$.

  2. ii.

Actual walls: Let $W_{v,w}$ be a numerical wall, defined by classes $v$, $w$
satisfying $u=v+w$. A subset $V\subset W_{v,w}$ is an actual wall if for each
point $\sigma\in V$, there is a short exact sequence

$0\to\mathcal{F}\to\mathcal{E}\to\mathcal{G}\to 0$

in $\mathcal{A}$ of $\sigma$-semistable objects
$\mathcal{F},\mathcal{E},\mathcal{G}$ with classes $u,v,w$ in $\Lambda$,
respectively, such that
$\lambda_{\sigma}(\mathcal{F})=\lambda_{\sigma}(\mathcal{E})=\lambda_{\sigma}(\mathcal{G})$.

When the context is clear, we drop the word “actual” and just say “wall”.

Given a union of walls, we refer to each connected component of its complement
in $\mathrm{Stab}_{\Lambda}(X)$ as a _chamber_. By the arguments in [7,
Section 9] there is a locally finite collection of (actual) walls in
$\mathrm{Stab}_{\Lambda}(X)$, each being a closed codimension one manifold
with boundary, such that the set of stable objects in $\mathcal{A}$ with class
$u\in\Lambda$ remains constant within each chamber, and there are no strictly
semistable objects in a chamber.

We say that a short exact sequence as in (ii) above _defines_ the wall.
Relaxing this, an unordered pair $\\{\mathcal{F},\mathcal{G}\\}$ _defines_ the
wall if there is a short exact sequence in either direction (i.e. we allow the
roles of sub and quotient objects to be swapped) as in (ii). Semistability of
$\mathcal{F}$ and $\mathcal{G}$ is automatic, i.e. it follows from
semistability of $\mathcal{E}$ and the equality between slopes, but see Remark
2.4.

######  Remark 2.3.

A _very weak_ stability condition $(\mathcal{A},Z)$ is a weakening of the
above concept (see Piyaratne–Toda [18]) where $Z$ is allowed to map nonzero
objects in $\mathcal{A}$ to zero. One may define an associated slope function
$\lambda$ as before, with the convention that $\lambda(\mathcal{E})=+\infty$
also when $Z(\mathcal{E})=0$. An object $\mathcal{E}\in\mathcal{A}$ is
declared to be stable if every nontrivial subobject
$\mathcal{F}\subsetneq\mathcal{E}$ satisfies
$\lambda(\mathcal{F})<\lambda(\mathcal{E}/\mathcal{F})$, and semistable when
nonstrict inequality is allowed. With this definition one avoids the need to
treat cases where $Z(\mathcal{F})=0$ or $Z(\mathcal{E}/\mathcal{F})=0$
separately. We will not need to go into detail.

###  2.3 Construction of stability conditions on threefolds

We next recall the “double tilt” construction of stability conditions by
Bayer–Macrì–Toda [5]. For this it is necessary to assume that the threefold
$X$ satisfies a certain “Bogomolov inequality” type condition [4, Conjecture
4.1]), which is known in several cases including $\mathbb{P}^{3}$ [14]. Fix a
polarization $H$ on $X$; on $\mathbb{P}^{3}$ this will be a (hyper)plane.

####  2.3.1 Slope stability

Let $\beta\in\mathbb{R}$. The twisted Chern character of a sheaf or a complex
$\mathcal{E}$ on $X$ is defined by $\mathrm{ch}^{\beta}(\mathcal{E})=e^{-\beta
H}\mathrm{ch}(\mathcal{E})$. Its homogeneous components are

$\displaystyle\mathrm{ch}_{0}^{\beta}(\mathcal{E})$
$\displaystyle=\mathrm{ch}_{0}(\mathcal{E})=\mathrm{rk}(\mathcal{E})$
$\displaystyle\mathrm{ch}_{1}^{\beta}(\mathcal{E})$
$\displaystyle=\mathrm{ch}_{1}(\mathcal{E})-\beta
H\mathrm{ch}_{0}(\mathcal{E})$
$\displaystyle\mathrm{ch}_{2}^{\beta}(\mathcal{E})$
$\displaystyle=\mathrm{ch}_{2}(\mathcal{E})-\beta
H\mathrm{ch}_{1}(\mathcal{E})+\frac{\beta^{2}H^{2}}{2}\mathrm{ch}_{0}(\mathcal{E})$
$\displaystyle\mathrm{ch}_{3}^{\beta}(\mathcal{E})$
$\displaystyle=\mathrm{ch}_{3}(\mathcal{E})-\beta
H\mathrm{ch}_{2}(\mathcal{E})+\frac{\beta^{2}H^{2}}{2}\mathrm{ch}_{1}(\mathcal{E})-\frac{\beta^{3}H^{3}}{6}\mathrm{ch}_{0}(\mathcal{E}).$

The twisted slope stability function on the abelian category $\mathrm{Coh}(X)$
of coherent sheaves is given by

$\mu_{\beta}(\mathcal{E})=\begin{cases}\frac{H^{2}ch_{1}^{\beta}(\mathcal{E})}{H^{3}ch_{0}^{\beta}(\mathcal{E})}&\text{if
$\mathrm{rk}(\mathcal{E})\neq 0$,}\\\ +\infty&\text{else.}\end{cases}$ (2.1)

This is the slope of a very weak stability condition. Notice that
$\mu_{\beta}(\mathcal{E})=\mu(\mathcal{E})-\beta$, where $\mu(\mathcal{E})$ is
the classical slope stability function. A sheaf
$\mathcal{E}\in\mathrm{Coh}(X)$ which is (semi)stable with respect to this
very weak stability condition is called $\mu_{\beta}$-(semi)stable (or slope
(semi)stable).

####  2.3.2 Tilt stability

Next, define the following full subcategories of $\mathrm{Coh}(X)$

$\displaystyle\mathcal{T}_{\beta}$
$\displaystyle=\\{\mathcal{E}\in\mathrm{Coh}(X)\;\vline\;\text{Any quotient
$\mathcal{E}\twoheadrightarrow\mathcal{G}$ satisfies
$\mu_{\beta}(\mathcal{G})>0$}\\},$ $\displaystyle\mathcal{T}^{\perp}_{\beta}$
$\displaystyle=\\{\mathcal{E}\in\mathrm{Coh}(X)\;\vline\;\text{Any subsheaf
$\mathcal{F}\subset\mathcal{E}$ satisfies $\mu_{\beta}(\mathcal{F})\leq
0$}\\}.$

The pair $(\mathcal{T}_{\beta},\mathcal{T}^{\perp}_{\beta})$ is a torsion pair
([7, Definition 3.2]) in $\mathrm{Coh}(X)$. Tilt the category
$\mathrm{Coh}(X)$ with respect to this torsion pair and denote the obtained
heart by
$\mathrm{Coh}^{\beta}(X)=\left<\mathcal{T}^{\perp}_{\beta}[1],\mathcal{T}_{\beta}\right>$.
Thus every object $\mathcal{E}\in\mathrm{Coh}^{\beta}(X)$ fits in a short
exact sequence

$0\to\mathcal{H}^{-1}(\mathcal{E})[1]\to\mathcal{E}\to\mathcal{H}^{0}(\mathcal{E})\to
0$ (2.2)

with $\mathcal{H}^{-1}(\mathcal{E})\in\mathcal{T}^{\perp}_{\beta}$ and
$\mathcal{H}^{0}(\mathcal{E})\in\mathcal{T}_{\beta}$.

Let $(\alpha,\beta)\in\mathbb{R}_{>0}\times\mathbb{R}$, and let

$Z^{\mathrm{tilt}}_{\alpha,\beta}(\mathcal{E})=-Hch_{2}^{\beta}(\mathcal{E})+\frac{\alpha^{2}}{2}H^{3}ch_{0}^{\beta}(\mathcal{E})+iH^{2}ch_{1}^{\beta}(\mathcal{E}).$
(2.3)

The associated slope function is

$\nu_{\alpha,\beta}(\mathcal{E})=\begin{cases}\frac{Hch_{2}^{\beta}(\mathcal{E})-\frac{\alpha^{2}}{2}H^{3}ch_{0}^{\beta}(\mathcal{E})}{H^{2}ch_{1}^{\beta}(\mathcal{E})}&\text{if
$H^{2}ch_{1}^{\beta}(\mathcal{E})\neq 0$,}\\\ +\infty&\text{else},\end{cases}$

(see [15, Section 9.1] for more details).

By [4, Proposition B.2 (case $B=\beta H$)], the pair
$(\mathrm{Coh}^{\beta}(X),Z^{\mathrm{tilt}}_{\alpha,\beta})$ is a very weak
stability condition continuously parametrized by
$(\alpha,\beta)\in\mathbb{R}_{>0}\times\mathbb{R}$. An object
$\mathcal{E}\in\mathrm{Coh}^{\beta}(X)$ which is (semi)stable with respect to
this very weak stability condition is called $\nu_{\alpha,\beta}$-(semi)stable
(or tilt (semi)stable). Moreover the parameter space
$\mathbb{R}_{>0}\times\mathbb{R}$ admits a wall and chamber structure [4,
Proposition B.5], in which walls are nested semicircles centered on the
$\beta$-axis, or vertical lines (we view $\alpha$ as the vertical axis) [20,
Theorem 3.3], and a numerical wall is either an actual wall everywhere or
nowhere. We refer to them as “tilt-stability walls” or “$\nu$-walls”
interchangeably.

######  Remark 2.4.

Walls in the parameter space for tilt stability are defined analogously to
Definition 2.2. With tilt-stability in mind we make the following observation:
let $\mathcal{E}$ be strictly semistable with respect to a very weak stability
condition $(\mathcal{A},Z)$. By definition this means that $\mathcal{E}$ is
semistable and there exists a short exact sequence

$0\to\mathcal{F}\to\mathcal{E}\to\mathcal{G}\to 0$

in $\mathcal{A}$ such that all three objects share the same slope. This
implies that $\mathcal{F}$ is semistable, since any destabilizing subobject of
$\mathcal{F}$ would also destabilize $\mathcal{E}$. When very weak stability
conditions are allowed, however, $\mathcal{G}$ may not be semistable: this
happens exactly when $\mathcal{G}$ has finite slope and there is a nontrivial
subobject $\mathcal{G}^{\prime}\subset\mathcal{G}$ such that
$Z(\mathcal{G}^{\prime})=0$. In this case let
$\mathcal{G}^{\prime}\subset\mathcal{G}$ be the maximal such subobject: then
$\mathcal{G}/\mathcal{G}^{\prime}$ is semistable (it is in fact the final
factor in the Harder–Narasimhan filtration of $\mathcal{G}$) and has the same
slope as $\mathcal{G}$. Moreover the kernel $\mathcal{F}^{\prime}$ of the
composite $\mathcal{E}\to\mathcal{G}\to\mathcal{G}/\mathcal{G}^{\prime}$ has
the same slope as $\mathcal{F}$. Thus in the short exact sequence

$0\to\mathcal{F}^{\prime}\to\mathcal{E}\to\mathcal{G}/\mathcal{G}^{\prime}\to
0$

all objects are semistable and of the same slope. So when looking for walls we
may as in Definition 2.2 assume all objects in the defining short exact
sequence to be semistable, even in the very weak situation.

The following is the Bogomolov inequality for tilt-stability:

######  Proposition 2.5.

[5, Corollary 7.3.2] Any $\nu_{\alpha,\beta}$-semistable object
$\mathcal{E}\in\mathrm{Coh}^{\beta}(X)$ satisfies

$\overline{\Delta}_{H}(\mathcal{E})=\big{(}H^{2}\mathrm{ch}^{\beta}_{1}(\mathcal{E})\big{)}^{2}-2H^{3}\mathrm{ch}^{\beta}_{0}(\mathcal{E})H\mathrm{ch}_{2}^{\beta}(\mathcal{E})\geq
0.$

####  2.3.3 Bridgeland stability

Define the following full subcategories of $\mathrm{Coh}^{\beta}(X)$:

$\displaystyle\mathcal{T}^{\prime}_{\alpha,\beta}$
$\displaystyle=\\{\mathcal{E}\in\mathrm{Coh}^{\beta}(X)\;\vline\;\text{Any
quotient $\mathcal{E}\twoheadrightarrow\mathcal{G}$ satisfies
$\nu_{\alpha,\beta}(\mathcal{G})>0$}\\},$
$\displaystyle\mathcal{T}^{{}^{\prime}\perp}_{\alpha,\beta}$
$\displaystyle=\\{\mathcal{E}\in\mathrm{Coh}^{\beta}(X)\;\vline\;\text{Any
subsheaf $\mathcal{F}\subset\mathcal{E}$ satisfies
$\nu_{\alpha,\beta}(\mathcal{F})\leq 0$}\\}.$

They form a torsion pair. Tilting $\mathrm{Coh}^{\beta}(X)$ with respect to
this pair yields stability conditions
$(\mathcal{A}_{\alpha,\beta}(X),Z_{\alpha,\beta,s})$ ([4, Theorem 8.6, Lemma
8.8]) on $X$, where
$\mathcal{A}_{\alpha,\beta}(X)=\left<\mathcal{T}^{{}^{\prime}\perp}_{\alpha,\beta}[1],\mathcal{T}^{\prime}_{\alpha,\beta}\right>$
and

$Z_{\alpha,\beta,s}=-\mathrm{ch}_{3}^{\beta}+\alpha^{2}(\frac{1}{6}+s)H^{2}\mathrm{ch}_{1}^{\beta}+i(H\mathrm{ch}_{2}^{\beta}-\frac{\alpha^{2}}{2}H^{3}\mathrm{ch}_{0}^{\beta}).$
(2.4)

The slope function of $Z_{\alpha,\beta,s}$ is given by

$\lambda_{\alpha,\beta,s}(\mathcal{E})=\frac{\mathrm{ch}_{3}^{\beta}(\mathcal{E})-\alpha^{2}(\frac{1}{6}+s)H^{2}\mathrm{ch}_{1}^{\beta}(\mathcal{E})}{H\mathrm{ch}_{2}^{\beta}(\mathcal{E})-\frac{\alpha^{2}}{2}H^{3}\mathrm{ch}_{0}^{\beta}(\mathcal{E})}$

with $\lambda_{\alpha,\beta,s}(\mathcal{E})=+\infty$ when
$H\mathrm{ch}_{2}^{\beta}(\mathcal{E})=\frac{\alpha^{2}}{2}H^{3}\mathrm{ch}_{0}^{\beta}(\mathcal{E})$.

An object $\mathcal{E}\in\mathcal{A}_{\alpha,\beta}(X)$ which is (semi)stable
with respect to this stability condition is called
$\lambda_{\alpha,\beta,s}$-(semi)stable.

The stability conditions $(\mathcal{A}_{\alpha,\beta}(X),Z_{\alpha,\beta,s})$
are continuously parametrized by
$(\alpha,\beta,s)\in\mathbb{R}_{>0}\times\mathbb{R}\times\mathbb{R}_{>0}$ ([4,
Proposition 8.10]). We refer to walls in
$\mathbb{R}_{>0}\times\mathbb{R}\times\mathbb{R}_{>0}$ as “$\lambda$-walls”.

The following lemma allows us to identify moduli spaces of slope-stable
sheaves with moduli spaces of tilt-stable sheaves, given the right conditions:

######  Lemma 2.6 ([11, Lemma 1.4]).

On $\mathbb{P}^{3}$, let $v=(v_{0},v_{1},v_{2},v_{3})\in\Lambda$ satisfy
$\mu_{\beta}(v)>0$ and assume $(v_{0},v_{1})$ is primitive. Then an object
$\mathcal{E}\in\mathrm{Coh}^{\beta}(\mathbb{P}^{3})$ with
$\mathrm{ch}(\mathcal{E})=v$ is $\nu_{\alpha,\beta}$-stable for all $\alpha\gg
0$ if and only if $\mathcal{E}$ is a slope stable sheaf.

###  2.4 Comparison between $\nu$-stability and $\lambda$-stability — after
Schmidt

Let $\mathcal{E}$ be an object in $D^{b}(X)$. Throughout this section, let
$(\alpha_{0},\beta_{0})\in\mathbb{R}_{>0}\times\mathbb{R}$ satisfy
$\nu_{\alpha_{0},\beta_{0}}(\mathcal{E})=0$, and fix $s>0$. We shall summarize
a series of results by Schmidt [20] enabling us to compare walls and chambers
with respect to $\nu$-stability with those of $\lambda$-stability. (Looking
ahead to our application illustrated in Figures 1 and 2, the dashed hyperbola
is the solution set to $\nu_{\alpha,\beta}(\mathcal{E})=0$.)

Consider the following conditions on $\mathcal{E}$:

  1. 1.

$\mathcal{E}$ is a $\nu_{\alpha_{0},\beta_{0}}$-stable object in
$\mathrm{Coh}^{\beta_{0}}(X)$.

  2. 2.

$\mathcal{E}$ is a $\lambda_{\alpha,\beta,s}$-stable object in
$\mathcal{A}^{\alpha,\beta}(X)$, for all $(\alpha,\beta)$ in an open
neighborhood of $(\alpha_{0},\beta_{0})$ with
$\nu_{\alpha,\beta}(\mathcal{E})>0$.

  3. 3.

$\mathcal{E}$ is a $\lambda_{\alpha,\beta,s}$-semistable object in
$\mathcal{A}^{\alpha,\beta}(X)$, for all $(\alpha,\beta)$ in an open
neighborhood of $(\alpha_{0},\beta_{0})$ with
$\nu_{\alpha,\beta}(\mathcal{E})>0$.

  4. 4.

$\mathcal{E}$ is a $\nu_{\alpha_{0},\beta_{0}}$-semistable object in
$\mathrm{Coh}^{\beta_{0}}(X)$.

Obviously there are implications (1) $\implies$ (4) and (2) $\implies$ (3).
The following says that, under a mild condition on $\mathrm{ch}(\mathcal{E})$,
there are in fact implications

$(1)\implies(2)\implies(3)\implies(4)$

so that $\lambda$-stability in a certain sense refines $\nu$-stability:

######  Theorem 2.7 (Schmidt).

The implication $(1)\implies(2)$ above always holds. If
$H^{2}\mathrm{ch}^{\beta_{0}}_{1}(\mathcal{E})>0$ and
$\overline{\Delta}_{H}(\mathcal{E})>0$ then also the implication
$(3)\implies(4)$ holds.

For the proof we refer to Schmidt [20]: the first implication follows from
Lemma 6.2 in _loc. cit._ and the second follows from Lemmas 6.3 and 6.4.

Schmidt furthermore compares walls for $\nu$-stability and
$\lambda$-stability, for objects $\mathcal{E}$ in some fixed class
$v\in\Lambda$. Let

$\mathcal{F}\to\mathcal{E}\to\mathcal{G}\to\mathcal{F}[1]$ (2.5)

be a triangle in $D^{b}(X)$ with $\mathcal{E}$ in class $v$.

  * •

Say that (2.5) _defines a $\nu$-wall_ through $(\alpha_{0},\beta_{0})$ if
$\mathcal{F}$,$\mathcal{E}$,$\mathcal{G}$ are
$\nu_{\alpha_{0},\beta_{0}}$-semistable objects in
$\mathrm{Coh}^{\beta_{0}}(X)$ and
$\nu_{\alpha_{0},\beta_{0}}(\mathcal{F})=\nu_{\alpha_{0},\beta_{0}}(\mathcal{G})$
(which is thus zero).

  * •

Say that (2.5) _defines a $\lambda$-wall at the $\nu$-positive side of
$(\alpha_{0},\beta_{0})$_ if there is an open neighborhood $U$ of
$(\alpha_{0},\beta_{0})$ such that, writing

$W=\\{(\alpha,\beta)\in U\;\mid\;\text{$\nu_{\alpha,\beta}(v)>0$ and
$\lambda_{\alpha,\beta,s}(\mathcal{F})=\lambda_{\alpha,\beta,s}(\mathcal{G})$}\\}$

the following holds: $(\alpha_{0},\beta_{0})$ is in the closure of $W$ and
$\mathcal{F},\mathcal{E},\mathcal{G}$ are
$\lambda_{\alpha,\beta,s}$-semistable objects in
$\mathcal{A}^{\alpha,\beta}(X)$ for all $(\alpha,\beta)\in W$.

Note that the assumption that $\mathcal{F},\mathcal{E},\mathcal{G}$ are all in
$\mathrm{Coh}^{\beta_{0}}(X)$ or in $\mathcal{A}^{\alpha,\beta}(X)$ implies
that the triangle (2.5) is a short exact sequence

$0\to\mathcal{F}\to\mathcal{E}\to\mathcal{G}\to 0$

in that abelian category.

######  Theorem 2.8 (Schmidt).

Let $\mathcal{E}$ be an object in $D^{b}(X)$ satisfying
$\overline{\Delta}_{H}(\mathcal{E})>0$ and let
$(\alpha_{0},\beta_{0})\in\mathbb{R}_{>0}\times\mathbb{R}$ such that
$\nu_{\alpha_{0},\beta_{0}}(\mathcal{E})=0$ and
$\mathrm{ch}^{\beta_{0}}_{1}(\mathcal{E})>0$.

  1. 1.

If a triangle (2.5) defines a $\lambda$-wall on the $\nu$-positive side of
$(\alpha_{0},\beta_{0})$, then it also defines a $\nu$-wall through
$(\alpha_{0},\beta_{0})$.

  2. 2.

Suppose a triangle (2.5) defines a $\nu$-wall through $(\alpha_{0},\beta_{0})$
and $\mathcal{F}$, $\mathcal{G}$ are $\nu_{\alpha_{0},\beta_{0}}$-stable.
Moreover let

$W=\\{(\alpha,\beta)\;\mid\;\text{$\nu_{\alpha,\beta}(v)>0$ and
$\lambda_{\alpha,\beta,s}(\mathcal{F})=\lambda_{\alpha,\beta,s}(\mathcal{G})$}\\}$

and suppose there are points $(\alpha,\beta)\in W$ arbitrarily close to
$(\alpha_{0},\beta_{0})$ such that $\nu_{\alpha,\beta}(\mathcal{F})>0$ and
$\nu_{\alpha,\beta}(\mathcal{G})>0$. Then (2.5) defines a $\lambda$-wall on
the $\nu$-positive side of $(\alpha_{0},\beta_{0})$, namely $W$.

For the proof we refer to Schmidt [20]: part (1) is Schmidt’s Theorem 6.1(1)
and part (2) is the special case $n=1$ of Schmidt’s Theorem 6.1(4). To align
the notation, in part (1) Schmidt’s $\mathcal{F},\mathcal{E},\mathcal{G}$ are
our $\mathcal{F}[1],\mathcal{E}[1],\mathcal{G}[1]$. To apply Theorem 6.1(1)
these are required to be $\lambda_{\alpha_{0},\beta_{0},s}$-semistable objects
in $\mathcal{A}^{\alpha_{0},\beta_{0}}(X)$; this is ensured by Schmidt’s Lemma
6.3.

To control how the set of stable objects changes as a $\lambda$-wall is
crossed, we take advantage of the fact that the $\lambda$-walls we obtain are
defined by short exact sequences with stable sub- and quotient objects (in
other words, only two Jordan–Hölder factors on the wall) and apply:

######  Proposition 2.9.

Suppose $\mathcal{F}$ and $\mathcal{G}$ are $\lambda_{\alpha,\beta,s}$-stable
objects in $\mathcal{A}^{\alpha,\beta}(X)$. Then there is a neighborhood $U$
of $(\alpha,\beta)$ such that for all $(\alpha^{\prime},\beta^{\prime})\in U$
and all nonsplit extensions

$0\to\mathcal{F}\to\mathcal{E}\to\mathcal{G}\to 0$

the object $\mathcal{E}$ is
$\lambda_{\alpha^{\prime},\beta^{\prime},s}$-stable if and only if
$\lambda_{\alpha^{\prime},\beta^{\prime},s}(\mathcal{F})<\lambda_{\alpha^{\prime},\beta^{\prime},s}(\mathcal{G})$.

This result is stated and proved (for arbitrary Bridgeland stability
conditions) in Schmidt [20, Lemma 3.11], and credited there also to
Bayer–Macrì [3, Lemma 5.9].

##  3 Wall and chamber structure

The starting point for the entire discussion that follows is a simple minded
observation. Namely, let $V\subset\mathbb{P}^{3}$ be a plane and let $Y$ be
the union of a conic in $V$ and a point $P$ also in $V$. Then there is a short
exact sequence

$0\to\mathcal{O}_{\mathbb{P}^{3}}(-1)\to\mathcal{I}_{Y}\to\mathcal{I}_{P/V}(-2)\to
0$ (3.1)

(read $\mathcal{O}_{\mathbb{P}^{3}}(-1)$ as the ideal of $V$ and
$\mathcal{I}_{P/V}(-2)$ as the relative ideal of $Y\subset V$). If we instead
let $Y$ be the union of a conic in $V$ and a point $P$ outside of $V$ then
there is a short exact sequence

$0\to\mathcal{I}_{P}(-1)\to\mathcal{I}_{Y}\to\mathcal{O}_{V}(-2)\to 0$ (3.2)

(read $\mathcal{I}_{P}(-1)$ as the ideal of $\\{P\\}\cup V$ and
$\mathcal{O}_{V}(-2)$ as the relative ideal of a conic in $V$). The claim is
that in a certain region of the stability manifold of $\mathbb{P}^{3}$, there
are exactly two walls with respect to the Chern character
$\mathrm{ch}(\mathcal{I}_{Y})=(1,0,-2,2)$, and they are defined precisely by
the two pairs of sub and quotient objects appearing in the short exact
sequences (3.1) and (3.2).

Mimicking Schmidt’s work for twisted cubics (and their deformations), we argue
via tilt stability. Since $\nu_{\alpha,\beta}$-stability only involves Chern
classes of codimension at least one, and the above two short exact sequences
are indistinguishable in codimension one, they give rise to one and the same
wall in the tilt stability parameter space. Making this precise is the content
of Section 3.1. Moving on to $\lambda_{\alpha,\beta,s}$-stability, we apply
Schmidt’s method to see that the single $\nu_{\alpha,\beta}$-wall “sprouts”
two distinct $\lambda_{\alpha,\beta,s}$-walls corresponding to (3.1) and
(3.2). This is carried out in Section 3.2.

###  3.1 $\nu$-stability wall

Throughout we specialize to $X=\mathbb{P}^{3}$ with $H$ a (hyper)plane. Let
$v=(1,0,-2,2)$ be the Chern character of ideal sheaves $\mathcal{I}_{Y}$ of
subschemes $Y\in\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$.

For $(\alpha,\beta)\in\mathbb{R}_{>0}\times\mathbb{R}$ we have the tilted
abelian category $\mathrm{Coh}^{\beta}(\mathbb{P}^{3})$ and the slope function
$\nu_{\alpha,\beta}$. We concentrate on the region $\beta<0$, in which any
ideal $\mathcal{I}_{Y}$ of a subscheme $Y\subset\mathbb{P}^{3}$ of dimension
$\leq 1$ satisfies

$\mu_{\beta}(\mathcal{I}_{Y})=\frac{c_{1}(\mathcal{I}_{Y})}{\mathrm{rk}(\mathcal{I}_{Y})}-\beta=-\beta>0.$

As $\mathcal{I}_{Y}$ is $\mu$-stable also $\mu_{\beta}(\mathcal{G})>0$ for
every quotient $\mathcal{I}_{Y}\twoheadrightarrow\mathcal{G}$ and so
$\mathcal{I}_{Y}\in\mathcal{T}_{\beta}$. In particular
$\mathcal{I}_{Y}\in\mathrm{Coh}^{\beta}(\mathbb{P}^{3})$.

We begin by establishing that there is exactly one tilt-stability wall in the
region $\beta<0$. The result as well as the argument is analogous to the
analysis for twisted cubics by Schmidt [20, Theorem 5.3], except that twisted
cubics come with a second wall that destabilizes all objects — for our skew
lines there is no such final wall.

$\beta$$\alpha$$W$$-\tfrac{5}{2}$$\tfrac{3}{2}$$\beta^{2}-\alpha^{2}=4$ Figure
1: The unique semicircular wall $W$ (solid) for $\nu$-stability and the
hyperbola (dashed) from Equation 3.3.

######  Proposition 3.1.

There is exactly one tilt-stability wall for objects with Chern character
$v=(1,0,-2,2)$ in the region $\beta<0$: it is the semicircle

$W\colon\quad\alpha^{2}+(\beta+\tfrac{5}{2})^{2}=(\tfrac{3}{2})^{2}.$

The wall is defined by exactly the unordered pairs of the following two types:

  1. (1)

$\left\\{\mathcal{I}_{P}(-1),\mathcal{O}_{V}(-2)\right\\}$, where
$V\subset\mathbb{P}^{3}$ is a plane and $P\in V$, and

  2. (2)

$\left\\{\mathcal{O}_{\mathbb{P}^{3}}(-1),\mathcal{I}_{P/V}(-2)\right\\}$,
where $V\subset\mathbb{P}^{3}$ is a plane and $P\not\in V$.

Moreover, the four sheaves figuring in the above unordered pairs are
$\nu_{\alpha,\beta}$-stable objects in $\mathrm{Coh}^{\beta}(\mathbb{P}^{3})$
for all $(\alpha,\beta)$ on $W$.

The wall $W$ and the hyperbola $\nu_{\alpha,\beta}(v)=0$, intersecting at
$(\alpha,\beta)=(3/2,-5/2)$, are shown in Figure 1. Note that we visualize the
$\alpha$-axis as the vertical one.

We first prove the final claim in the proposition. Here is a slightly more
general statement:

######  Lemma 3.2.

  1. 1.

Let $Z\subset\mathbb{P}^{3}$ be a finite, possibly empty subscheme. Then the
ideal sheaf $\mathcal{I}_{Z}(-1)$ is a $\nu_{\alpha,\beta}$-stable object in
$\mathrm{Coh}^{\beta}(\mathbb{P}^{3})$ for all $\alpha>0$ and $\beta<-1$.

  2. 2.

Let $V\subset\mathbb{P}^{3}$ be a plane and $Z\subset V$ be a finite, possibly
empty subscheme. Then the relative ideal sheaf $\mathcal{I}_{Z/V}(-2)$ is a
$\nu_{\alpha,\beta}$-stable object in $\mathrm{Coh}^{\beta}(\mathbb{P}^{3})$
for all $(\alpha,\beta)\in\mathbb{R}_{>0}\times\mathbb{R}$ such that

$\alpha^{2}+(\beta+\tfrac{5}{2})^{2}>(\tfrac{1}{2})^{2}.$

######  Remark 3.3.

The condition on $(\alpha,\beta)$ in part (2) is necessary because of a wall
for $\mathcal{I}_{Z/V}(-2)$. For simplicity let $Z$ be empty. There is a short
exact sequence of coherent sheaves

$0\to\mathcal{O}_{\mathbb{P}^{3}}(-3)\to\mathcal{O}_{\mathbb{P}^{3}}(-2)\to\mathcal{O}_{V}(-2)\to
0$

which yields a short exact sequence

$0\to\mathcal{O}_{\mathbb{P}^{3}}(-2)\to\mathcal{O}_{V}(-2)\to\mathcal{O}_{\mathbb{P}^{3}}(-3)[1]\to
0$

in $\mathrm{Coh}^{\beta}(\mathbb{P}^{3})$ when $-3<\beta<-2$. The condition
$\nu_{\alpha,\beta}(\mathcal{O}_{\mathbb{P}^{3}}(-2))<\nu_{\alpha,\beta}(\mathcal{O}_{V}(-2))$
is exactly the inequality in (2).

###### Proof of Lemma 3.2.

The sheaf $\mathcal{I}_{Z}(-1)$ is $\mu$-stable and satisfies
$\mu_{\beta}(\mathcal{I}_{Z}(-1))=-1-\beta$. For all $\beta<-1$ it is thus an
object in $\mathcal{T}_{\beta}$ and so also in
$\mathrm{Coh}^{\beta}(\mathbb{P}^{3})$. Since $\mathcal{I}_{Z/V}(-2)$ is a
torsion sheaf it too belongs to $\mathcal{T}_{\beta}$ and so to
$\mathrm{Coh}^{\beta}(\mathbb{P}^{3})$, for all $\beta$.

We reduce to the situation $Z=\emptyset$. First consider $\mathcal{I}_{Z}(-1)$
and assume $\beta<-1$. Note that $\mathcal{I}_{Z}(-1)$ is a subobject of
$\mathcal{O}_{\mathbb{P}^{3}}(-1)$ also in
$\mathrm{Coh}^{\beta}(\mathbb{P}^{3})$ since the torsion sheaf
$\mathcal{O}_{Z}$ belongs to that category and hence

$0\to\mathcal{I}_{Z}(-1)\to\mathcal{O}_{\mathbb{P}^{3}}(-1)\to\mathcal{O}_{Z}\to
0$

is a short exact sequence in $\mathrm{Coh}^{\beta}(\mathbb{P}^{3})$. Suppose
$\mathcal{O}_{\mathbb{P}^{3}}(-1)$ is $\nu_{\alpha,\beta}$-stable. Let
$\mathcal{F}\subset\mathcal{I}_{Z}(-1)$ be a proper nonzero subobject in
$\mathrm{Coh}^{\beta}(\mathbb{P}^{3})$ with quotient $\mathcal{G}$. View
$\mathcal{F}$ also as a subobject of $\mathcal{O}_{\mathbb{P}^{3}}(-1)$, with
quotient $\mathcal{G}^{\prime}$. Then $\nu_{\alpha,\beta}$ cannot distinguish
between $\mathcal{G}$ and $\mathcal{G}^{\prime}$. Thus if
$\mathcal{O}_{\mathbb{P}^{3}}(-1)$ is $\nu_{\alpha,\beta}$-stable then

$\nu_{\alpha,\beta}(\mathcal{F})<\nu_{\alpha,\beta}(\mathcal{G}^{\prime})=\nu_{\alpha,\beta}(\mathcal{G})$

and so $\mathcal{I}_{Z}(-1)$ is $\nu_{\alpha,\beta}$-stable as well. The
reduction from $\mathcal{I}_{Z/V}(-2)$ to $\mathcal{O}_{V}(-2)$ is completely
analogous.

$\nu_{\alpha,\beta}$-stability of the line bundle
$\mathcal{O}_{\mathbb{P}^{3}}(-1)$ is a consequence of
$\overline{\Delta}_{H}(\mathcal{O}_{\mathbb{P}^{3}}(-1))=0$, by [5,
Proposition 7.4.1].

The main task is to establish $\nu_{\alpha,\beta}$-stability of
$\mathcal{O}_{V}(-2)$ in the region defined in part (2). By point (3) of [20,
Theorem 3.3], the ray $\beta=-\frac{5}{2}$ intersects all potential
semicircular $\nu$-walls for $\mathrm{ch}(\mathcal{O}_{V}(-2))$ at their top
point, meaning they must be centered at $(0,-\frac{5}{2})$. All such
semicircles of radius bigger than $\frac{1}{2}$ will intersect the ray
$\beta=-2$ (as well as $\beta=-3$). Thus it suffices to prove that
$\mathcal{O}_{V}(-2)$ is $\nu_{\alpha,\beta}$-stable for all $\alpha>0$ and
all _integers_ $\beta$.

For such $(\alpha,\beta)$, suppose

$0\to\mathcal{F}\to\mathcal{O}_{V}(-2)\to\mathcal{G}\to 0$

is a short exact sequence in $\mathrm{Coh}^{\beta}(\mathbb{P}^{3})$ with
$\mathcal{F}\neq 0$. We claim that $\mathrm{ch}_{1}^{\beta}(\mathcal{G})=0$.
This yields the result, since then $\nu_{\alpha,\beta}(\mathcal{G})=\infty$
and so $\mathcal{O}_{V}(-2)$ is $\nu_{\alpha,\beta}$-stable.

Let $r_{\mathcal{F}}=H^{3}\mathrm{ch}_{0}(\mathcal{F})$ and
$c_{\mathcal{F}}=H^{2}\mathrm{ch}_{1}(\mathcal{F})$, i.e. the rank and first
Chern class considered as integers. Also let
$r_{\mathcal{G}}=H^{3}\mathrm{ch}_{0}(\mathcal{G})$ and
$c_{\mathcal{G}}=H^{2}\mathrm{ch}_{1}(\mathcal{G})$. By the short exact
sequence we have

$r_{\mathcal{F}}+r_{\mathcal{G}}=0\quad\text{and}\quad
c_{\mathcal{F}}+c_{\mathcal{G}}=1.$

The induced long exact cohomology sequence of sheaves shows that
$\mathcal{H}^{-1}(\mathcal{F})=0$, so from the short exact sequence

$0\to\mathcal{H}^{-1}(\mathcal{F})[1]\to\mathcal{F}\to\mathcal{H}^{0}(\mathcal{F})\to
0$

we see that $\mathcal{F}\cong\mathcal{H}^{0}(\mathcal{F})$ is a coherent sheaf
in $\mathcal{T}_{\beta}$. The remaining long exact sequence is

$0\to\mathcal{H}^{-1}(\mathcal{G})\to\mathcal{F}\to\mathcal{O}_{V}(-2)\to\mathcal{H}^{0}(\mathcal{G})\to
0$

The map into $\mathcal{F}$ cannot be an isomorphism, since
$\mathcal{H}^{-1}(\mathcal{G})$ is in $\mathcal{T}^{\perp}_{\beta}$ and
$\mathcal{F}$ is in $\mathcal{T}_{\beta}$ and is nonzero by assumption.
Therefore the map in the middle is nonzero and so the rightmost sheaf
$\mathcal{H}^{0}(\mathcal{G})$ is a proper quotient of $\mathcal{O}_{V}(-2)$
and so is a torsion sheaf supported in dimension $\leq 1$. Thus only
$\mathcal{H}^{-1}(\mathcal{G})$ contributes to $r_{\mathcal{G}}$ and
$c_{\mathcal{G}}$.

Suppose $r_{\mathcal{F}}\neq 0$. As $\mathcal{F}\in\mathcal{T}_{\beta}$ and
$\mathcal{H}^{-1}(\mathcal{G})\in\mathcal{T}^{\perp}_{\beta}$ we have

$\begin{cases}\mu_{\beta}(\mathcal{F})>0\\\
\mu_{\beta}(\mathcal{H}^{-1}(\mathcal{G}))\leq
0\end{cases}\implies\begin{cases}\dfrac{c_{\mathcal{F}}}{r_{\mathcal{F}}}-\beta>0\\\
\dfrac{c_{\mathcal{G}}}{r_{\mathcal{G}}}-\beta\leq
0\end{cases}\implies\begin{cases}\dfrac{c_{\mathcal{G}}-1}{r_{\mathcal{G}}}-\beta>0\\\
\dfrac{c_{\mathcal{G}}}{r_{\mathcal{G}}}-\beta\leq 0\end{cases}$

and since $r_{\mathcal{G}}=-r_{\mathcal{F}}$ is negative we get $0\leq
c_{\mathcal{G}}-\beta r_{\mathcal{G}}<1$. Since these are integers we must
have $c_{\mathcal{G}}-\beta r_{\mathcal{G}}=0$. Thus

$\mathrm{ch}_{1}^{\beta}(\mathcal{G})=(c_{\mathcal{G}}-\beta
r_{\mathcal{G}})H=0$

as claimed.

If on the other hand $r_{\mathcal{F}}=0$ then also
$\mathcal{H}^{-1}(\mathcal{G})$ has rank zero and hence must be zero as there
are no torsion sheaves in $\mathcal{T}^{\perp}_{\beta}$. Thus also
$\mathcal{G}=\mathcal{H}^{0}(\mathcal{G})$ is a sheaf, with vanishing rank and
first Chern class. Again $\mathrm{ch}_{1}^{\beta}(\mathcal{G})=0$ as claimed.
This completes the proof. ∎

By explicit computation (see [20, Theorem 3.3]), all numerical tilt walls with
respect to $v=(1,0,-2,2)$ in the region $\beta<0$ are nested semicircles. More
precisely, each is centered on the axis $\alpha=0$ and has top point on the
curve $\nu_{\alpha,\beta}(v)=0$, that is the hyperbola

$\beta^{2}-\alpha^{2}=4.$ (3.3)

In particular every tilt wall must intersect the ray $\beta=-2$.

We establish in the following lemma that there is at most one tilt stability
wall intersecting the ray $\beta=-2$ for Chern character $v$ and $\beta<0$. We
also give the possible Chern characters of sub- and quotient objects that
define it. This lemma is tightly analogous to Schmidt [20, Lemma 5.5]. We use
an asterisk $*$ to denote an unspecified numerical value.

######  Lemma 3.4.

Let $\beta_{0}=-2$ and let $\alpha>0$ be arbitrary. Suppose there is a short
exact sequence

$0\to\mathcal{F}\to\mathcal{E}\to\mathcal{G}\to 0$

of $\nu_{\alpha,\beta_{0}}$-semistable objects in
$\mathrm{Coh}^{\beta_{0}}(\mathbb{P}^{3})$ with
$\mathrm{ch}(\mathcal{E})=(1,0,-2,*)$ and
$\nu_{\alpha,\beta_{0}}(\mathcal{F})=\nu_{\alpha,\beta_{0}}(\mathcal{G})$.
Then

$\mathrm{ch}^{\beta_{0}}(\mathcal{F})=(1,1,\tfrac{1}{2},*)\quad\text{and}\quad\mathrm{ch}^{\beta_{0}}(\mathcal{G})=(0,1,-\tfrac{1}{2},*)$

or the other way around.

###### Proof.

Keep $\beta_{0}=-2$ throughout. We compute
$\mathrm{ch}^{\beta_{0}}(\mathcal{E})=(1,2,0,*)$. Let
$\mathrm{ch}^{\beta_{0}}(\mathcal{F})=(r,c,d,*)$ with $r,c\in\mathbb{Z}$ and
$d\in\frac{1}{2}\mathbb{Z}$. Then
$\mathrm{ch}^{\beta_{0}}(\mathcal{G})=(1-r,2-c,-d,*)$.

Since the (very weak) stability function $Z^{\textrm{tilt}}$ sends effective
classes to the upper half plane $\mathbb{H}\cup\\{0\\}$ and
$Z^{\textrm{tilt}}(\mathcal{E})=Z^{\textrm{tilt}}(\mathcal{F})+Z^{\textrm{tilt}}(\mathcal{G})$
we have

$0\leq\Im Z^{\textrm{tilt}}(\mathcal{F})\leq\Im
Z^{\textrm{tilt}}(\mathcal{E}).$

Since $\Im Z^{\textrm{tilt}}=H\mathrm{ch}_{1}^{\beta_{0}}$ this gives $0\leq
c\leq 2$.

If $c=0$ then $\nu_{\alpha,\beta_{0}}(\mathcal{F})=\infty$ and
$\nu_{\alpha,\beta_{0}}(\mathcal{G})<\infty$, which is a contradiction.
Similarly if $c=2$ then $\nu_{\alpha,\beta_{0}}(\mathcal{F})<\infty$ and
$\nu_{\alpha,\beta_{0}}(\mathcal{G})=\infty$, again a contradiction. Therefore
$c=1$.

With $c=1$ we compute

$\nu_{\alpha,\beta_{0}}(\mathcal{F})=d-\tfrac{1}{2}\alpha^{2}r\quad\text{and}\quad\nu_{\alpha,\beta_{0}}(\mathcal{G})=-d-\tfrac{1}{2}\alpha^{2}(1-r)$

and so the condition
$\nu_{\alpha,\beta_{0}}(\mathcal{F})=\nu_{\alpha,\beta_{0}}(\mathcal{G})$ says

$\alpha^{2}=\frac{4d}{2r-1}$ (3.4)

so this expression must be strictly positive.

Suppose $r\geq 1$ and apply the Bogomolov inequality (Proposition 2.5) to
$\mathcal{F}$:

$0\leq\overline{\Delta}_{H}(\mathcal{F})=1-2rd\quad\implies\quad
d\leq\frac{1}{2r}$

When $r\geq 1$ this gives $d\leq\frac{1}{2}$. On the other hand the positivity
of (3.4) gives $d>0$ and as $d$ is a half integer this leaves only the
possibility $d=\frac{1}{2}$ and $r=1$.

Similarly suppose $r\leq 0$ and apply the Bogomolov inequality to
$\mathcal{G}$:

$0\leq\overline{\Delta}_{H}(\mathcal{G})=1+2(1-r)d\quad\implies\quad
d\geq-\frac{1}{2(1-r)}$

When $r\leq 0$ this gives $d\geq-\tfrac{1}{2}$. On the other hand the
positivity of (3.4) gives $d<0$ and as $d$ is a half integer this leaves only
the possibility $d=-\frac{1}{2}$ and $r=0$. ∎

###### Proof of Proposition 3.1.

Assume there is a tilt stability wall for $v=(1,0,-2,2)$, i.e. there is a
short exact sequence

$0\to\mathcal{F}\to\mathcal{E}\to\mathcal{G}\to 0$

of $\nu_{\alpha,\beta}$-semistable objects in
$\mathrm{Coh}^{\beta}(\mathbb{P}^{3})$ with
$\mathrm{ch}(\mathcal{E})=(1,0,-2,2)$ and
$\nu_{\alpha,\beta}(\mathcal{F})=\nu_{\alpha,\beta}(\mathcal{G})$. As already
pointed out, the same conditions then hold for some $(\alpha,\beta_{0})$ with
$\beta_{0}=-2$. Then by Lemma 3.4, up to swapping $\mathcal{F}$ and
$\mathcal{G}$, we have

$\displaystyle\mathrm{ch}^{\beta_{0}}(\mathcal{F})$
$\displaystyle=(1,1,\tfrac{1}{2},*)$ (3.5)
$\displaystyle\mathrm{ch}^{\beta_{0}}(\mathcal{G})$
$\displaystyle=(0,1,-\tfrac{1}{2},*).$ (3.6)

Given any pair $\mathcal{F},\mathcal{G}$ of such objects, write out the
condition $\nu_{\alpha,\beta}(\mathcal{F})=\nu_{\alpha,\beta}(\mathcal{G})$ on
$(\alpha,\beta)$ to obtain the equation for the wall in question; this yields
the semicircle as claimed. Thus we have proved that there is at most one tilt-
wall and found its equation.

A further result of Schmidt [20, Lemma 5.4] (which requires $\beta$ to be
integral, and so applies for $\beta_{0}=-2$) says that the only
$\nu_{\alpha,\beta_{0}}$-semistable objects in
$\mathrm{Coh}^{\beta_{0}}(\mathbb{P}^{3})$ with the invariants (3.5) and (3.6)
are

$\displaystyle\mathcal{F}$ $\displaystyle\cong\mathcal{I}_{Z}(-1)$
$\displaystyle\mathcal{G}$ $\displaystyle\cong\mathcal{I}_{Z^{\prime}/V}(-2)$

for a finite subscheme $Z\subset\mathbb{P}^{3}$, a plane
$V\subset\mathbb{P}^{3}$ and a finite subscheme $Z^{\prime}\subset V$ (where
$Z$ and $Z^{\prime}$ are allowed to be empty). Let $n$ and $n^{\prime}$ denote
the lengths of $Z$ and $Z^{\prime}$, respectively. Again for $\beta_{0}=-2$ we
compute

$\displaystyle\mathrm{ch}^{\beta_{0}}_{3}(\mathcal{F})$
$\displaystyle=\mathrm{ch}^{\beta_{0}}_{3}(\mathcal{I}_{Z}(-1))=\tfrac{1}{6}-n$
$\displaystyle\mathrm{ch}^{\beta_{0}}_{3}(\mathcal{G})$
$\displaystyle=\mathrm{ch}^{\beta_{0}}_{3}(\mathcal{I}_{Z^{\prime}/V}(-2))=\tfrac{1}{6}-n^{\prime}$

and moreover $\mathrm{ch}^{\beta_{0}}_{3}(\mathcal{E})=-\frac{2}{3}$. Thus
from
$\mathrm{ch}^{\beta_{0}}_{3}(\mathcal{E})=\mathrm{ch}^{\beta_{0}}_{3}(\mathcal{F})+\mathrm{ch}^{\beta_{0}}_{3}(\mathcal{G})$
we find

$n+n^{\prime}=1$

and so either $Z$ is empty and $Z^{\prime}$ is a point, or $Z$ is a point and
$Z^{\prime}$ is empty. This proves that only the two listed pairs of
semistable objects $\mathcal{F},\mathcal{G}$ may occur in a short exact
sequence defining the wall.

To finish the proof it only remains to show that both pairs of objects listed
do in fact realize the wall. By Lemma 3.2, the sheaves $\mathcal{I}_{Z}(-1)$
and $\mathcal{I}_{Z^{\prime}/V}(-2)$ are in
$\mathrm{Coh}^{\beta}(\mathbb{P}^{3})$ and are $\nu_{\alpha,\beta}$-semistable
(in fact $\nu_{\alpha,\beta}$-stable) for all $(\alpha,\beta)$ on the
semicircle. Also, the ideal $\mathcal{E}=\mathcal{I}_{Y}$ of any
$Y\in\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$ is an object in
$\mathrm{Coh}^{\beta}(\mathbb{P}^{3})$ (when $\beta<0$) and since any ideal is
$\mu$-stable it is $\nu_{\alpha,\beta}$-stable for $\alpha\gg 0$ (by
Proposition 2.6). Hence it is $\nu_{\alpha,\beta}$-stable outside the
semicircle and at least $\nu_{\alpha,\beta}$-semistable on the semicircle.
Thus, short exact sequences of the types (3.1) and (3.2) define the wall and
we are done. ∎

###  3.2 $\lambda$-stability walls

Next we apply Schmidt’s Theorem 2.8 to the single $\nu_{\alpha,\beta}$-wall
found in Proposition 3.1; this yields two $\lambda_{\alpha,\beta,s}$-walls.

We set up notation first: for
$(\alpha,\beta,s)\in\mathbb{R}_{>0}\times\mathbb{R}\times\mathbb{R}_{>0}$ we
have the doubly tilted category $\mathcal{A}^{\alpha,\beta}(\mathbb{P}^{3})$
and the slope function $\lambda_{\alpha,\beta,s}$. Once and for all we fix an
arbitrary value $s>0$ and view the $(\alpha,\beta)$-plane
$\mathbb{R}_{>0}\times\mathbb{R}$ as parametrizing both
$\nu_{\alpha,\beta}$-stability and $\lambda_{\alpha,\beta,s}$-stability; as
before we restrict to $\beta<0$. Walls and chambers are taken with respect to
the Chern character $v=(1,0,-2,2)$.

Write $P_{v}\subset\mathbb{R}_{>0}\times\mathbb{R}$ for the open subset
defined by $\nu_{\alpha,\beta}(v)>0$ and $\beta<0$; this is the region to the
left of the hyperbola (3.3) in Figure 2. Theorem 2.8 addresses walls in
$P_{v}$ close to the boundary hyperbola.

######  Proposition 3.5.

There are exactly two $\lambda_{\alpha,\beta,s}$-walls with respect to
$v=(1,0,-2,2)$ in $P_{v}$ whose closure intersect the hyperbola (3.3). They
are defined exactly by the two pairs of objects listed in Proposition 3.1.

This means that the two walls are

$W_{1}=\\{(\alpha,\beta)\in
P_{v}\;\vline\;\lambda_{\alpha,\beta,s}(\mathcal{O}_{\mathbb{P}^{3}}(-1))=\lambda_{\alpha,\beta,s}(\mathcal{I}_{Q/V}(-2))\\}$
(3.7)

and

$W_{2}=\\{(\alpha,\beta)\in
P_{v}\;\vline\;\lambda_{\alpha,\beta,s}(\mathcal{I}_{P}(-1))=\lambda_{\alpha,\beta,s}(\mathcal{O}_{V}(-2))\\}$
(3.8)

and the pair of objects defining each wall (close to $(\alpha_{0},\beta_{0})$)
is unique.

We refrain from writing out the (quartic) equations defining them. They do
depend on $s$, but independently of $s$ they both intersect the hyperbola
(3.3) in $(\alpha_{0},\beta_{0})=(\frac{3}{2},-\frac{5}{2})$ and as we will
show in the following proof, $W_{1}$ has negative slope at
$(\alpha_{0},\beta_{0})$ whereas $W_{2}$ has positive slope there. Thus
$W_{1}$ lies above $W_{2}$ ($\alpha$ bigger) in the intersection between
$P_{v}$ and a small open neighborhood of $(\alpha_{0},\beta_{0})$.

###### Proof.

We apply Schmidt’s Theorem 2.8. Firstly, when $\mathrm{ch}(\mathcal{E})=v$ we
have $\mathrm{ch}_{1}^{\beta}(\mathcal{E})=v_{1}-\beta v_{0}=-\beta>0$ and
$\overline{\Delta}_{H}(v)=v_{1}^{2}-2v_{0}v_{2}=4>0$ so the theorem applies.
The first part of the Theorem says that any $\lambda$-wall in $P_{v}$, having
a point $(\alpha_{0},\beta_{0})$ with $\nu_{\alpha_{0},\beta_{0}}(v)=0$ in its
closure, must be defined by one of the two pairs $(\mathcal{F},\mathcal{G})$
listed in Proposition 3.1. This leaves $W_{1}$ and $W_{2}$ as the only
candidates. Moreover the sub- and quotient objects $\mathcal{F}$ and
$\mathcal{G}$ appearing are $\nu_{\alpha_{0},\beta_{0}}$-stable by Lemma 3.2.
Thus the second part of the theorem says that conversely, $W_{1}$ and $W_{2}$
are indeed $\lambda$-walls, provided they contain points $(\alpha,\beta)$
arbitrarily close to $(\alpha_{0},\beta_{0})=(\frac{3}{2},-\frac{5}{2})$ such
that $\nu_{\alpha,\beta}(\mathcal{F})>0$ and
$\nu_{\alpha,\beta}(\mathcal{G})>0$. It remains to check this last condition.

So let $(\mathcal{F},\mathcal{G})$ be one of the pairs
$(\mathcal{O}_{\mathbb{P}^{3}}(-1),\mathcal{I}_{P/V}(-2))$ or
$(\mathcal{I}_{P}(-1),\mathcal{O}_{V}(-2))$. The region $P_{v}$ is bounded by
the hyperbola $\nu_{\alpha,\beta}(v)=0$ and implicit differentiation readily
shows that this has slope $\frac{d\alpha}{d\beta}=-5/3$ at
$(\alpha_{0},\beta_{0})$. Similarly $\nu_{\alpha,\beta}(\mathcal{F})=0$ has
slope $-1$ and $\nu_{\alpha,\beta}(\mathcal{G})=0$ is just the line
$\beta=-5/2$, and in each case $\nu_{\alpha,\beta}>0$ is the region to the
left of these boundary curves. Thus it suffices to show that our walls have
slope $>-1$ at $(\alpha_{0},\beta_{0})$. Now each wall $W_{i}$ is defined by
the condition
$\lambda_{\alpha,\beta,s}(\mathcal{F})=\lambda_{\alpha,\beta,s}(\mathcal{G})$,
which is equivalent to

$(\Re Z(\mathcal{F}))(\Im Z(\mathcal{G}))=(\Re Z(\mathcal{G}))(\Im
Z(\mathcal{F}))$ (3.9)

where $Z=Z_{\alpha,\beta,s}$ is the stability function defined in (2.4) from
Section 2.3. Implicit differentiation of this equation at
$(\alpha_{0},\beta_{0})$ gives, after some work, that $W_{1}$ has slope

$-\left(\frac{27s}{16}+1\right)^{-1}\in(-1,0)$ (3.10)

and $W_{2}$ has slope

$\left(\frac{27s}{4}+1\right)^{-1}\in(0,1)$ (3.11)

both of which are $>-1$, and we are done. ∎

$W_{2}$$W_{1}$IIIIII$\beta$$\alpha$$-\tfrac{5}{2}$$\tfrac{3}{2}$$\beta^{2}-\alpha^{2}=4$
Figure 2: The two walls $W_{1}$ and $W_{2}$ (solid) for $\lambda$-stability
separating three chambers, together with the hyperbola (dashed) from Equation
3.3.

We are now in position to prove Theorem 1.1. By Proposition 3.5 there exists
an open (connected) neighborhood $N\subset\mathbb{R}_{>0}\times\mathbb{R}$
around the $\beta<0$ branch of the hyperbola (3.3), such that the only
$\lambda$-walls in $N\cap P_{v}$ are $W_{1}$ and $W_{2}$, defined in (3.7) and
(3.8). Moreover it follows from the slopes (3.10) and (3.11) that, after
shrinking $N$ further if necessary, $W_{1}$ lies above ($\alpha$ bigger)
$W_{2}$ throughout $N\cap P_{v}$. Thus the two walls separate $N$ into three
chambers, which we label I, II and III in order of decreasing $\alpha$.

###### Proof of Theorem 1.1 (I).

By Proposition 3.1 there is a single semicircular wall (in the region
$\beta<0$) for $\nu$-stability. It follows from Theorem 2.7 that the class of
$\lambda_{\alpha,\beta,s}$-stable objects in
$\mathcal{A}^{\alpha,\beta}(\mathbb{P}^{3})$ for $(\alpha,\beta)$ in chamber I
coincides with the class of $\nu_{\alpha,\beta}$-stable objects in
$\mathrm{Coh}^{\beta}(\mathbb{P}^{3})$ for $(\alpha,\beta)$ outside the single
$\nu$-wall (up to shrinking $N$ even further if necessary).

Moreover, for $\alpha$ sufficiently big, the $\nu$-stable objects in
$\mathrm{Coh}^{\beta}(\mathbb{P}^{3})$ are exactly the $\mu$-stable coherent
sheaves (Proposition 2.6). For Chern character $v=(1,0,-2,2)$ these are the
ideals $\mathcal{I}_{Y}$ with $Y\in\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$. ∎

###### Proof of Theorem 1.1 (II).

Let $\mathcal{E}$ be $\lambda_{\alpha,\beta,s}$-stable for $(\alpha,\beta)$ in
chamber II. Since semistability is a closed property, $\mathcal{E}$ is
semistable on the wall $W_{1}$. If $\mathcal{E}$ is stable on the wall, then
it is also stable in chamber I hence it is an ideal sheaf in
$\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$ by part (I). Such an ideal remains
stable on the wall if and only if it is not an extension of the type (3.1),
that is if and only if it is the ideal of a nonplanar subscheme. This is case
(II)(i) in the Theorem.

If on the other hand $\mathcal{E}$ is stable in chamber II, but strictly
semistable on $W_{1}$, then by Proposition 3.5 it is a nonsplit extension of
the pair

$\big{(}\mathcal{O}_{\mathbb{P}^{3}}(-1),\mathcal{I}_{P/V}(-2)\big{)}$ (3.12)

and we determine the direction of the extension (which object is the subobject
and which is the quotient) as follows: we claim that

$\lambda_{\alpha,\beta,s}(\mathcal{I}_{P/V}(-2))<\lambda_{\alpha,\beta,s}(\mathcal{O}_{\mathbb{P}^{3}}(-1))$
(3.13)

for all $(\alpha,\beta)$ in chamber II sufficiently close to
$(\alpha_{0},\beta_{0})$. Granted this, it follows that for $\mathcal{E}$ to
be stable in chamber II it must be a nonsplit extension as in case (II)(ii) in
the Theorem. Conversely it follows from Proposition 2.9 that every such
nonsplit extension is indeed stable in chamber II. To verify (3.13) we let

$\Phi(\alpha,\beta)=\Re Z(\mathcal{F})\Im Z(\mathcal{G})-\Re Z(\mathcal{G})\Im
Z(\mathcal{F})$ (3.14)

with $Z=Z_{\alpha,\beta,s}$, $\mathcal{F}=\mathcal{O}_{\mathbb{P}^{3}}(-1)$
and $\mathcal{G}=\mathcal{I}_{P/V}(-2)$. Thus $W_{1}$ is defined by
$\Phi(\alpha,\beta)=0$ and (3.13) is equivalent to $\Phi(\alpha,\beta)<0$. It
thus suffices to check that the partial derivative of $\Phi$ with respect to
$\alpha$ is positive at $(\alpha_{0},\beta_{0})$. An explicit computation
yields in fact

$\frac{\partial\Phi}{\partial\alpha}(\alpha_{0},\beta_{0})=2+\frac{27s}{8}>0.$

It remains only to show uniqueness of the nonsplit extensions
$\mathcal{F}_{P,V}$, that is

$\dim\mathrm{Ext}^{1}_{\mathbb{P}^{3}}(\mathcal{O}_{\mathbb{P}^{3}}(-1),\mathcal{I}_{P/V}(-2))=1.$

But this space is $H^{1}(\mathcal{I}_{P/V}(-1))$, which is isomorphic to
$H^{0}(k(P))=k$ via the short exact sequence

$0\to\mathcal{I}_{P/V}(-1)\to\mathcal{O}_{V}(-1)\to k(P)\to 0.$

∎

###### Proof of Theorem 1.1 (III).

Let $\mathcal{E}$ be $\lambda_{\alpha,\beta,s}$-stable for $(\alpha,\beta)$ in
chamber III. Since semistability is a closed property, $\mathcal{E}$ is
semistable on the wall $W_{2}$. If $\mathcal{E}$ is stable on $W_{2}$, then it
is stable in chamber II. This means two things: first, by part (II) of the
Theorem $\mathcal{E}$ is either an ideal sheaf of a nonplanar subscheme or a
nonsplit extension $\mathcal{F}_{P,V}$ as in case (II)(ii). Second, to remain
stable on $W_{2}$ the object $\mathcal{E}$ cannot be in a short exact sequence
of the type (3.2) ruling out ideal sheaves of plane conics union a point. Also
the sheaves $\mathcal{F}_{P,V}$ sit in short exact sequences of this type, as
we show in Lemma 3.6 below (the vertical short exact sequence in the middle),
and so are ruled out as well. Hence $\mathcal{E}$ is an ideal sheaf of a
disjoint pair of lines as claimed in (III)(i).

If on the other hand $\mathcal{E}$ is strictly semistable on $W_{2}$, then by
Proposition 3.5 $\mathcal{E}$ is a nonsplit extension (in either direction) of
the pair

$\big{(}\mathcal{I}_{P}(-1),\mathcal{O}_{V}(-2)\big{)}.$

Now we claim that

$\lambda_{\alpha,\beta,s}(\mathcal{O}_{V}(-2)<\lambda_{\alpha,\beta,s}(\mathcal{I}_{P}(-1))$

for all $(\alpha,\beta)$ in chamber III sufficiently close to
$(\alpha_{0},\beta_{0})$. We prove this as in part II above, by partial
differentiation of $\Phi$ defined in (3.14), this time with
$\mathcal{F}=\mathcal{I}_{P}(-1)$ and $\mathcal{G}=\mathcal{O}_{V}(-2)$. We
find

$\frac{\partial\Phi}{\partial\alpha}(\alpha_{0},\beta_{0})=\frac{1}{2}+\frac{27s}{8}>0.$

As before we conclude that $\mathcal{E}$ is a nonsplit extension as in
(III)(ii) and by Proposition 2.9 all such extensions are stable.

It remains to verify uniqueness of the extensions $\mathcal{G}_{P,V}$, i.e.

$\dim\mathrm{Ext}^{1}_{\mathbb{P}^{3}}(\mathcal{I}_{P}(-1),\mathcal{O}_{V}(-2))=1$

when $P\in V$. For this first apply
$\operatorname{Hom}(-,\mathcal{O}_{V}(-1))$ to the short exact sequence

$0\to\mathcal{I}_{P}\to\mathcal{O}_{\mathbb{P}^{3}}\to k(P)\to 0$

to obtain a long exact sequence which together with the vanishing of
$H^{1}(\mathcal{O}_{V}(-1))$ and $H^{2}(\mathcal{O}_{V}(-1))$ gives an
isomorphism

$\mathrm{Ext}^{1}_{\mathbb{P}^{3}}(\mathcal{I}_{P},\mathcal{O}_{V}(-1))\cong\mathrm{Ext}^{2}(k(P),\mathcal{O}_{V}(-1))$

and ignoring twists, as these are not seen by $k(P)$, the right hand side is
Serre dual to $\mathrm{Ext}^{1}(\mathcal{O}_{V},k(P))$. This is one
dimensional as is seen by applying $\operatorname{Hom}(-,k(P))$ to the
sequence

$0\to\mathcal{O}_{\mathbb{P}^{3}}(-1)\to\mathcal{O}_{\mathbb{P}^{3}}\to\mathcal{O}_{V}\to
0.$

∎

###  3.3 The special sheaves

Let $\mathcal{F}=\mathcal{F}_{P,V}$ and $\mathcal{G}=\mathcal{G}_{P,V}$ denote
sheaves given by nonsplit extensions of the form (1.1) and (1.2),
respectively. The definition through (unique) nonsplit extensions is indirect
and it is useful to have alternative constructions available. We give such
constructions here and compute the spaces of first order infinitesimal
deformations.

######  Lemma 3.6.

There is a commutative diagram with exact rows and columns as follows:

${0}$${0}$${\mathcal{I}_{P}(-1)}$${\mathcal{I}_{P}(-1)}$${0}$${\mathcal{I}_{P/V}(-2)}$${\mathcal{F}}$${\mathcal{O}_{\mathbb{P}^{3}}(-1)}$${0}$${0}$${\mathcal{I}_{P/V}(-2)}$${\mathcal{O}_{V}(-2)}$${k(P)}$${0}$${0}$${0}$

###### Proof.

Up to identifying the skyscraper sheaf $k(P)$ with any of its twists, there
are canonical short exact sequences as in the bottom row and the rightmost
column. The diagram can then be completed by letting $\mathcal{F}$ be the
fiber product as laid out by the square in the bottom right corner. It remains
only to verify that the middle row is nonsplit. But if it were split the
middle column twisted by $\mathcal{O}_{\mathbb{P}^{3}}(1)$ would be a short
exact sequence of the form

$0\to\mathcal{I}_{P}\to\mathcal{I}_{P/V}(-1)\oplus\mathcal{O}_{\mathbb{P}^{3}}\to\mathcal{O}_{V}(-1)\to
0.$

Taking global sections this yields a contradictory left exact sequence in
which all terms vanish except $H^{0}(\mathcal{O}_{\mathbb{P}^{3}})=k$. ∎

######  Proposition 3.7.

We have $\dim\mathrm{Ext}^{1}(\mathcal{F},\mathcal{F})=11$.

###### Proof.

We will actually only prove that the dimension is at most $11$. The opposite
inequality may be shown by similar techniques, although it follows from
viewing $\mathrm{Ext}^{1}(\mathcal{F},\mathcal{F})$ as a Zariski tangent space
to the $11$-dimensional moduli space $\mathcal{M}^{\mathrm{II}}$ studied in
the next section.

Apply $\operatorname{Hom}(-,\mathcal{F})$ to the middle row in the diagram in
Lemma 3.6. This yields a long exact sequence

$\cdots\to
H^{1}(\mathcal{F}(1))\to\mathrm{Ext}^{1}(\mathcal{F},\mathcal{F})\to\mathrm{Ext}^{1}(\mathcal{I}_{P/V}(-2),\mathcal{F})\to
H^{2}(\mathcal{F}(1))\to\cdots$

and from the middle column of the diagram we compute
$H^{1}(\mathcal{F}(1))=H^{2}(\mathcal{F}(1))=0$. Thus we proceed to show that
$\dim\mathrm{Ext}^{1}(\mathcal{I}_{P/V}(-2),\mathcal{F})\leq 11$.

Apply $\operatorname{Hom}(\mathcal{I}_{P/V}(-2),-)$ to the middle row in the
diagram. This yields a long exact sequence:

$\cdots\to\mathrm{Ext}^{1}(\mathcal{I}_{P/V},\mathcal{I}_{P/V})\to\mathrm{Ext}^{1}(\mathcal{I}_{P/V}(-2),\mathcal{F})\to\mathrm{Ext}^{1}(\mathcal{I}_{P/V}(-1),\mathcal{O}_{\mathbb{P}^{3}})\to\cdots$

The space on the right is Serre dual to $H^{2}(\mathcal{I}_{P/V}(-5))\cong
H^{2}(\mathcal{O}_{V}(-5))$, which again on $V$ is Serre dual to
$H^{0}(\mathcal{O}_{V}(2))$. This has dimension $6$. At least heuristically,
the space on the left should have dimension $5$, as it may be viewed as a
tangent space to the incidence variety
$I\subset\mathbb{P}^{3}\times\check{\mathbb{P}}^{3}$ seen as a moduli space
for the sheaves $\mathcal{I}_{P/V}$. More directly we may apply
$\operatorname{Hom}(\mathcal{I}_{P/V},-)$ to the Koszul complex on $V$

$0\to\mathcal{O}_{V}(-2)\to\mathcal{O}_{V}(-1)^{\oplus
2}\to\mathcal{I}_{P/V}\to 0$

to obtain a long exact sequence

$\cdots\to\underbrace{\mathrm{Ext}^{1}(\mathcal{I}_{P/V},\mathcal{O}_{V}(-1))^{\oplus
2}}_{2\cdot
2}\to\mathrm{Ext}^{1}(\mathcal{I}_{P/V},\mathcal{I}_{P/V})\to\underbrace{\mathrm{Ext}^{2}(\mathcal{I}_{P/V},\mathcal{O}_{V}(-2))}_{1}\to\cdots$

where the indicated dimensions may be computed by applying
$\operatorname{Hom}(\mathcal{I}_{P/V}(d),-)$ (for $d=1,2$) to the sequence

$0\to\mathcal{O}_{\mathbb{P}^{3}}(-1)\to\mathcal{O}_{\mathbb{P}^{3}}\to\mathcal{O}_{V}\to
0.$

We skip further details. It follows then that
$\dim\mathrm{Ext}^{1}(\mathcal{I}_{P/V},\mathcal{I}_{P/V})\leq 5$ and so
$\dim\mathrm{Ext}^{1}(\mathcal{I}_{P/V}(-2),\mathcal{F})$ is at most $5+6=11$.
∎

######  Lemma 3.8.

There is a commutative diagram with exact rows and columns as follows:

${0}$${0}$${\mathcal{O}_{\mathbb{P}^{3}}(-2)}$${\mathcal{O}_{\mathbb{P}^{3}}(-2)}$${0}$${\mathcal{O}_{V}(-2)}$${\mathcal{G}}$${\mathcal{I}_{P}(-1)}$${0}$${0}$${\mathcal{O}_{V}(-2)}$${\Omega^{1}_{V}}$${\mathcal{I}_{P/V}(-1)}$${0}$${0}$${0}$

###### Proof.

From the Euler sequence

$0\to\Omega^{1}_{V}\to\mathcal{O}_{V}(-1)^{\oplus 3}\to\mathcal{O}_{V}\to 0$
(3.15)

on $V\cong\mathbb{P}^{2}$ it follows that $\Omega^{1}_{V}(2)$ has a unique
section (up to scale) vanishing at $P\in V$. This leads to the short exact
sequence in the bottom row. Moreover there is a canonical short exact sequence
as in the rightmost column. The rest of the diagram can then be formed by
taking $\mathcal{G}$ to be the fiber product as laid out by the bottom right
square. It just remains to verify that the middle row is indeed nonsplit. But
if it were split the middle column would be a short exact sequence of the form

$0\to\mathcal{O}_{\mathbb{P}^{3}}(-2)\to\mathcal{O}_{V}(-2)\oplus\mathcal{I}_{P}(-1)\to\Omega^{1}_{V}\to
0.$

This sequence implies that $H^{1}(\Omega^{1}_{V}(-1))$ is isomorphic to
$H^{1}(\mathcal{I}_{P}(-2))$, which is one dimensional. But the Euler sequence
shows that in fact $H^{1}(\Omega^{1}_{V}(-1))=0$. ∎

######  Proposition 3.9.

We have $\dim\mathrm{Ext}^{1}(\mathcal{G},\mathcal{G})=8$.

###### Proof.

We will be using the short exact sequence

$0\to\mathcal{O}_{\mathbb{P}^{3}}(-2)\to\mathcal{G}\to\Omega^{1}_{V}\to 0$
(3.16)

which sits as the middle column in Lemma 3.8. As preparation we observe that
all (dimensions of) $H^{i}(\Omega^{1}_{V}(d))$ may be computed from the Euler
sequence, and this enables us to compute several $H^{i}(\mathcal{G}(d))$ from
(3.16). We use these results freely below without writing out further details.

Apply $\operatorname{Hom}(-,\mathcal{G})$ to (3.16) to produce a long exact
sequence

$\displaystyle 0$
$\displaystyle\to\underbrace{\operatorname{Hom}(\Omega^{1}_{V},\mathcal{G})}_{0}\to\underbrace{\operatorname{Hom}(\mathcal{G},\mathcal{G})}_{1}\to\underbrace{H^{0}(\mathcal{G}(2))}_{4}$
$\displaystyle\to\mathrm{Ext}^{1}(\Omega_{V}^{1},\mathcal{G})\to\mathrm{Ext}^{1}(\mathcal{G},\mathcal{G})\to\underbrace{H^{1}(\mathcal{G}(2))}_{0}\to\cdots$

in which we have indicated some of the dimensions: $H^{i}(\mathcal{G}(2))$ are
computed from (3.16) as sketched above, and since $\mathcal{G}$ is simple we
have $\operatorname{Hom}(\mathcal{G},\mathcal{G})=k$. For the same reason
(3.16) is nonsplit, which implies
$\operatorname{Hom}(\Omega^{1}_{V},\mathcal{G})=0$. It thus remains to see
that the dimension of $\mathrm{Ext}^{1}(\Omega^{1}_{V},\mathcal{G})$ is $11$.

Next apply $\operatorname{Hom}(-,\mathcal{G})$ to the Euler sequence. This
gives a long exact sequence

$\displaystyle\cdots$
$\displaystyle\to\underbrace{\operatorname{Hom}(\Omega^{1}_{V},\mathcal{G})}_{0}\to\underbrace{\mathrm{Ext}^{1}(\mathcal{O}_{V},\mathcal{G})}_{1}\to\underbrace{\mathrm{Ext}^{1}(\mathcal{O}_{V},\mathcal{G}(1))^{\oplus
3}}_{4\cdot 3}$
$\displaystyle\to\mathrm{Ext}^{1}(\Omega^{1}_{V},\mathcal{G})\to\underbrace{\mathrm{Ext}^{2}(\mathcal{O}_{V},\mathcal{G})}_{0}\to\cdots$

where again dimensions have been indicated: the vanishing of
$\operatorname{Hom}(\Omega^{1}_{V},\mathcal{G})$ has already been noted, and
there remain several spaces of the form
$\mathrm{Ext}^{i}(\mathcal{O}_{V},\mathcal{G}(d))$. These may be computed from
$H^{i}(\mathcal{G}(d))$ and the long exact sequence resulting from applying
$\operatorname{Hom}(-,\mathcal{G}(d))$ to

$0\to\mathcal{O}_{\mathbb{P}^{3}}(-1)\to\mathcal{O}_{\mathbb{P}^{3}}\to\mathcal{O}_{V}\to
0.$

It follows that $\dim\mathrm{Ext}^{1}(\Omega^{1}_{V},\mathcal{G})=11$ and we
are done. ∎

##  4 Moduli spaces and universal families

By the classification of stable objects in chamber II, the moduli space
$\mathcal{M}^{\mathrm{II}}$ is at least obtained as a set from
$\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})=\mathcal{C}\cup\mathcal{S}$ by just
replacing the divisor $E\subset\mathcal{C}$, parametrizing conics union a
point inside a plane, with the incidence variety $I$, parametrizing just pairs
$(P,V)$ of a point $P$ inside a plane $V$. Similarly, the moduli space
$\mathcal{M}^{\mathrm{III}}$ of stable objects in chamber III is obtained from
$\mathcal{M}^{\mathrm{II}}$ set-theoretically by removing
$\mathcal{M}^{\mathrm{II}}\setminus\mathcal{S}$ and replacing the divisor
$F\subset\mathcal{S}$, parametrizing pairs of incident lines with a spatial
embedded point at the intersection, with the incidence variety $I$. We shall
carry out each of these replacements as a contraction, i.e. a blow-down, and
prove that this indeed yields $\mathcal{M}^{\mathrm{II}}$ and
$\mathcal{M}^{\mathrm{III}}$, essentially by writing down a universal family
for each case.

###  4.1 The contraction $\mathcal{C}\to\mathcal{C}^{\prime}$

Recall that $\mathcal{C}$ is isomorphic to the blow-up of
$\mathbb{P}^{3}\times\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})$ along the universal
curve $\mathcal{Z}$, where $\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})$ is the
Hilbert scheme of plane conics in $\mathbb{P}^{3}$ [13]. The exceptional
divisor $E^{\prime}$ is comprised of plane conics with an embedded point.

It is helpful to keep an eye at the following diagram

${\mathcal{C}}$${\mathbb{P}^{3}\times\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})}$${\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})}$${\mathbb{P}^{3}}$${\check{\mathbb{P}}^{3}}$$\scriptstyle{b}$$\scriptstyle{\mathit{pr}_{1}}$$\scriptstyle{\mathit{pr}_{2}}$$\scriptstyle{\pi}$
(4.1)

where $\pi$ sends a conic $C\in\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})$ to the
plane $V\in\check{\mathbb{P}}^{3}$ it spans and $b$ is the blowup along the
universal family $\mathcal{Z}$ of conics.

######  Remark 4.1.

It will sometimes be useful to resort to explicit computation in local
coordinates. For this let $U\subset\check{\mathbb{P}}^{3}$ be the affine open
subset of planes $V\subset\mathbb{P}^{3}$ with equation of the form

$x_{3}=c_{0}x_{0}+c_{1}x_{1}+c_{2}x_{2}.$ (4.2)

Furthermore the $\mathbb{P}^{5}$ of symmetric $3\times 3$ matrices $(s_{ij})$
parametrizes plane conics

$\sum_{0\leq i,j\leq 2}s_{ij}x_{i}x_{j}=0$ (4.3)

so that
$\left.{\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})}\right|_{U}\cong\mathbb{P}^{5}\times
U$ with universal family defined by the two equations (4.2) and (4.3). This is
also the center for the blowup $b$, and we note that it is nonsingular.

######  Lemma 4.2.

$\pi$ is a Zariski locally trivial $\mathbb{P}^{5}$-bundle. More precisely,
let $I\subset\mathbb{P}^{3}\times\check{\mathbb{P}}^{3}$ be the incidence
variety and let

$\mathcal{E}=\mathit{pr}_{2*}(\left.{\mathit{pr}_{1}^{*}\mathcal{O}_{\mathbb{P}^{3}}(2)}\right|_{I}).$

Then $\mathcal{E}$ is locally free of rank $6$ and
$\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})\cong\mathbb{P}(\mathcal{E}^{\vee})$ over
$\check{\mathbb{P}}^{3}$.

Here $\mathbb{P}(\mathcal{E}^{\vee})$ denotes the projective bundle
parametrizing lines in the fibers of $\mathcal{E}$. Starting with the
observation that the fiber of $\mathcal{E}$ over $V\in\check{\mathbb{P}}^{3}$
is $H^{0}(V,\mathcal{O}_{V}(2))$ (note that $H^{1}(V,\mathcal{O}_{V}(2))=0$,
so base change in cohomology applies) the Lemma is straight forward and we
refrain from writing out details.

Now let $E\subset\mathcal{C}$ be the locus of planar $Y\in\mathcal{C}$. The
condition on a disjoint union $Y=C\cup\\{P\\}$ to be in $E$ is just that $P$
is in the plane $V$ spanned by $C$. For a conic with an embedded point the
condition $Y\subset V$ also singles out the scheme structure at the embedded
point. View $E$ as a variety over the incidence variety
$I\subset\mathbb{P}^{3}\times\check{\mathbb{P}}^{3}$ via the morphism
$(\mathit{id}_{\mathbb{P}^{3}}\times\pi)\circ b$ (refer to Figure 3 for simple
illustrations of the types of elements in $E$ and $E^{\prime}$. In fact, we
find it helpful for keeping track of the various divisors introduced in this
section.)

######  Proposition 4.3.

$E$ is a Zariski locally trivial $\mathbb{P}^{5}$-bundle over the incidence
variety $I\subset\mathbb{P}^{3}\times\check{\mathbb{P}}^{3}$. The restriction
$\left.{\mathcal{O}_{\mathcal{C}}(E)}\right|_{\mathbb{P}^{5}}$ to a fiber is
isomorphic to $\mathcal{O}_{\mathbb{P}^{5}}(-1)$.

Figure 3: A Circle represents a conic contained in a plane that is shown as a
parallelogram, and a red dot is a point, possibly embedded in the conic. The
arrow is the direction vector at an embedded point. Note that in the left
illustration, the arrow is strictly contained in the plane.

Before giving the proof, we harvest our application:

######  Corollary 4.4.

There exist a smooth algebraic space $\mathcal{C}^{\prime}$, a morphism
$\phi\colon\mathcal{C}\to\mathcal{C}^{\prime}$ and a closed embedding
$I\subset\mathcal{C}^{\prime}$, such that $\phi$ restricts to an isomorphism
from $\mathcal{C}\setminus E$ to $\mathcal{C^{\prime}}\setminus I$ and to the
given projective bundle structure $E\to I$. Moreover $\phi$ is the blowup of
$\mathcal{C}^{\prime}$ along $I$.

It is well known that the condition verified in Proposition 4.3 implies the
contractibility of $E/I$ in the above sense. In the category of analytic
spaces this is the Moishezon [16] or Fujiki–Nakano [17, 10] criterion. In the
category of algebraic spaces the contractibility is due to Artin [2, Corollary
6.11], although the statement there lacks the identification with a blowup.
Lascu [12, Théorème 1] however shows that once the contracted space
$\mathcal{C}^{\prime}$ as well as the image $I\subset\mathcal{C}^{\prime}$ of
the contracted locus are both smooth, it does follow that the contracting
morphism is a blowup. Strictly speaking Lascu works in the category of
varieties, but our $\mathcal{C}^{\prime}$ turns out to be a variety anyway:

######  Remark 4.5.

The algebraic space $\mathcal{C}^{\prime}$ is in fact a projective variety. We
prove this in Section 5 using Mori theory. As the arguments there and in the
present section are largely independent we separate the statements.

We also point out that the smooth contracted space $\mathcal{C}^{\prime}$ is
unique once it exists: in general, suppose $\phi\colon X\to U$ and $\psi\colon
X\to V$ are proper birational morphisms between irreducible separated
algebraic spaces (say, of finite type over $k$) with $U$ and $V$ normal.
Moreover assume that $\phi(x_{1})=\phi(x_{2})$ if and only if
$\psi(x_{1})=\psi(x_{2})$. Let $\Gamma$ denote the image of $(\phi,\psi)\colon
X\to U\times V$. Then each of the projections from $\Gamma$ to $U$ and $V$ is
birational and bijective and hence an isomorphism by Zariski’s Main Theorem
(for this in the language of algebraic spaces we refer to the Stacks Project
[21, Tag 05W7]).

###### Proof of Proposition 4.3..

Consider the divisor

$\overline{E}=\\{(P,C)\in\mathbb{P}^{3}\times\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})\;\vline\;\text{$P$
in the plane spanned by $C$}\\}.$

It follows from Lemma 4.2 that

$\mathbb{P}^{3}\times\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})\xrightarrow{id_{\mathbb{P}^{3}}\times\pi}\mathbb{P}^{3}\times\check{\mathbb{P}}^{3}$

is a $\mathbb{P}^{5}$-bundle, hence its restriction to
$\big{(}id_{\mathbb{P}^{3}}\times\pi\big{)}^{-1}(I)=\overline{E}$ is a
$\mathbb{P}^{5}$-bundle over
$I\subset\mathbb{P}^{3}\times\check{\mathbb{P}}^{3}$.

Now, $E$ is the strict transform of $\overline{E}$, i.e. its blow-up along
$\mathcal{Z}\subset\overline{E}$. But this is a Cartier divisor, since
$\overline{E}$ is smooth, and so $E\cong\overline{E}$. This proves the first
claim.

Again using that $\overline{E}$ is smooth, its strict transform $E$ satisfies
the linear equivalence

$E=b^{*}(\overline{E})-E^{\prime}.$ (4.4)

The term
$b^{*}(\overline{E})=b^{*}(\mathit{id}_{\mathbb{P}^{3}}\times\pi)^{*}(I)$ is a
pullback from the base of the $\mathbb{P}^{5}$-bundle, so its restriction to
any fiber is trivial. Thus it suffices to see that $E^{\prime}$ restricted to
a fiber $\mathbb{P}^{5}$ is a hyperplane. Now the isomorphism $b\colon
E\cong\overline{E}$ identifies $E\cap E^{\prime}\subset E$ with
$\mathcal{Z}\subset E$. In the local coordinates from Remark 4.1 the divisor
$E$ is given by equation (4.2) and $\mathcal{Z}$ is given by the additional
equation (4.3). Here $(s_{ij})$ are the coordinates on the fiber
$\mathbb{P}^{5}$ and clearly (4.3) defines a hyperplane in each fiber — it is
the linear condition on the space of plane conics given by passage through a
given point. ∎

######  Remark 4.6.

The locus $\mathcal{C}\cap\mathcal{S}$ consists of pairs of intersecting lines
with a _spatial_ embedded point at the intersection (and, as degenerate cases,
planar double lines with a spatial embedded point). On the other hand $E$
consists only of planar objects, so $E$ is disjoint from $\mathcal{S}$. Thus
we may extend the contraction $\phi$ to a morphism between algebraic spaces

$(\phi\cup\mathit{id})\colon\mathcal{C}\cup\mathcal{S}\to\mathcal{C}^{\prime}\cup\mathcal{S}$

which is an isomorphism away from $E$ and restricts to the
$\mathbb{P}^{5}$-bundle $E\to I$ as before.

###  4.2 Moduli in chamber II

In this section we shall modify the universal family on the Hilbert scheme
$\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})=\mathcal{C}\cup\mathcal{S}$ in such a
way that we replace its fibers over $E\subset\mathcal{C}$ with the objects
$\mathcal{F}_{P,V}$ in Theorem 1.1. This family induces a morphism

$\mathcal{C}\cup\mathcal{S}\to\mathcal{M}^{\mathrm{II}}$

and we conclude via uniqueness of normal (in this case smooth) contractions
that $\mathcal{M}^{\mathrm{II}}$ coincides with
$\mathcal{C}^{\prime}\cup\mathcal{S}$.

Here is the construction: let

$\mathcal{Y}\subset\mathbb{P}^{3}\times\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$

be the universal family and let

$\mathcal{V}\subset\mathbb{P}^{3}\times E$

be the $E$-flat family whose fiber $\mathcal{V}_{\xi}\subset\mathbb{P}^{3}$
over a point $\xi$ mapping to $(P,V)\in I$ is the plane $V$. Clearly
$\mathcal{V}$ can be written down as a pullback of the universal plane over
$\check{\mathbb{P}}^{3}$. We view $\mathcal{V}$ as a closed subscheme of
$\mathbb{P}^{3}\times\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$. Then our modified
universal family is the ideal sheaf
$\mathcal{I}_{\mathcal{Y}\cup\mathcal{V}}$.

######  Remark 4.7.

We emphasize the (to us, at least) unusual situation that
$\mathcal{I}_{\mathcal{Y}\cup\mathcal{V}}$ is the ideal of a very nonflat
subscheme, yet as we show below it is flat as a coherent sheaf. Its fibers
over points in $E$ are not ideals at all, but rather the objects
$\mathcal{F}_{P,V}$.

######  Theorem 4.8.

As above let $\mathcal{Y}$ be the universal family over
$\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$ and $\mathcal{V}$ the family of planes
in $\mathbb{P}^{3}$ parametrized by $E$.

  * (i)

$\mathcal{I}_{\mathcal{Y}\cup\mathcal{V}}$ is flat as a coherent sheaf over
$\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$. Its fibers
$\mathcal{I}_{\mathcal{Y}\cup\mathcal{V}}\otimes k(\xi)$ over
$\xi\in\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$ are stable objects for stability
conditions in chamber II.

  * (ii)

The morphism
$\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})\to\mathcal{M}^{\mathrm{II}}$ determined
by $\mathcal{I}_{\mathcal{Y}\cup\mathcal{V}}$ induces an isomorphism
$\mathcal{C}^{\prime}\cup\mathcal{S}\cong\mathcal{M}^{\mathrm{II}}$.

We begin by showing that the fibers
$\mathcal{I}_{\mathcal{Y}\cup\mathcal{V}}\otimes k(\xi)$ over $\xi\in E$ sit
in a short exact sequence of the type (1.1). The mechanism producing such a
short exact sequence is quite general. Note that when $Y=\mathcal{Y}_{\xi}$ is
a conic with a (possibly embedded) point $P$ in a plane $V=\mathcal{V}_{\xi}$,
we have $\mathcal{I}_{Y/V}\cong\mathcal{I}_{P/V}(-2)$ and
$\mathcal{I}_{V}\cong\mathcal{O}_{\mathbb{P}^{3}}(-1)$.

######  Lemma 4.9.

Let $X$ be a projective scheme, $\mathcal{Y}\subset X\times S$ an $S$-flat
subscheme, $E\subset S$ a Cartier divisor and $\mathcal{V}\subset E\times S$
an $E$-flat subscheme such that $\mathcal{Y}\cap(E\times
S)\subset\mathcal{V}$. Let $\xi\in E$. Then there is a short exact sequence

$0\to\mathcal{I}_{\mathcal{Y}_{\xi}/\mathcal{V}_{\xi}}\to\mathcal{I}_{\mathcal{Y}\cup\mathcal{V}}\otimes
k(\xi)\to\mathcal{I}_{\mathcal{V}_{\xi}}\to 0.$

In particular if $S$ is integral in a neighborhood of $E$ then
$\mathcal{I}_{\mathcal{Y}\cup\mathcal{V}}$ is flat over $S$.

###### Proof.

Observe that the last claim is implied by the first: outside of $E$ the ideal
$\mathcal{I}_{\mathcal{Y}\cup\mathcal{V}}$ agrees with
$\mathcal{I}_{\mathcal{Y}}$, which is flat. For $\xi\in E$ we have
$\mathcal{Y}_{\xi}\subset\mathcal{V}_{\xi}$ and so a short exact sequence

$0\to\mathcal{I}_{\mathcal{V}_{\xi}}\to\mathcal{I}_{\mathcal{Y}_{\xi}}\to\mathcal{I}_{\mathcal{Y}_{\xi}/\mathcal{V}_{\xi}}\to
0.$

The short exact sequence in the statement has the same sub and quotient
objects in opposite roles, so the Hilbert polynomial of
$\mathcal{I}_{\mathcal{Y}\cup\mathcal{V}}\otimes k(\xi)$ agrees with that of
$\mathcal{I}_{\mathcal{Y}_{\xi}}$. Thus the Hilbert polynomial of the fibers
of $\mathcal{I}_{\mathcal{Y}\cup\mathcal{V}}$ is constant; over an integral
base this implies flatness.

Begin with the short exact sequence

$0\to\mathcal{I}_{\mathcal{Y}\cup\mathcal{V}}\to\mathcal{O}_{X\times
S}\to\mathcal{O}_{\mathcal{Y}\cup\mathcal{V}}\to 0$

and tensor with $\mathcal{O}_{X\times E}$ to obtain the exact sequence

$0\to\mathcal{T}\\!\mathit{or}_{1}^{X\times
S}(\mathcal{O}_{\mathcal{Y}\cup\mathcal{V}},\mathcal{O}_{X\times
E})\to\left.{\mathcal{I}_{\mathcal{Y}\cup\mathcal{V}}}\right|_{E}\to\mathcal{O}_{X\times
E}\to\mathcal{O}_{\mathcal{V}}\to 0.$ (4.5)

The kernel of the rightmost map is clearly the ideal
$\mathcal{I}_{\mathcal{V}}\subset\mathcal{O}_{X\times E}$. To compute the Tor-
sheaf on the left use the short exact sequence

$0\to\mathcal{O}_{S}(-E)\to\mathcal{O}_{S}\to\mathcal{O}_{E}\to 0.$

Pull this back to $X\times S$ and tensor with
$\mathcal{O}_{\mathcal{Y}\cup\mathcal{V}}$ to see that
$\mathcal{T}\\!\mathit{or}_{1}^{X\times
S}(\mathcal{O}_{\mathcal{Y}\cup\mathcal{V}},\mathcal{O}_{X\times E})$ is
isomorphic to the kernel of the homomorphism

$\mathcal{O}_{\mathcal{Y}\cup\mathcal{V}}(-\mathit{pr}_{2}^{*}E)\to\mathcal{O}_{\mathcal{Y}\cup\mathcal{V}}$

which locally is multiplication by an equation for $E$. Thus

$\mathcal{T}\\!\mathit{or}_{1}^{X\times
S}(\mathcal{O}_{\mathcal{Y}\cup\mathcal{V}},\mathcal{O}_{X\times
E})\cong\mathcal{J}(-\mathit{pr}_{2}^{*}E)$

where $\mathcal{J}\subset\mathcal{O}_{\mathcal{Y}\cup\mathcal{V}}$ is the
ideal locally consisting of elements annihilated by an equation for $E$.

We compute $\mathcal{J}$ in an open affine subset $\operatorname{Spec}A\subset
X\times S$ in which $\mathcal{Y}$ and $\mathcal{V}$ are given by ideals
$I_{\mathcal{Y}}$ and $I_{\mathcal{V}}$ respectively and $f\in A$ is (the
pullback of) a local equation for $E$. Thus $\mathcal{J}$ corresponds to
$(I_{\mathcal{Y}}\cap I_{\mathcal{V}}:f)/(I_{\mathcal{Y}}\cap
I_{\mathcal{V}})$. Now $f\in I_{\mathcal{V}}$ since $\mathcal{V}\subset
X\times E$. This implies that for $g\in A$ the condition $fg\in
I_{\mathcal{Y}}$ is equivalent to $fg\in I_{\mathcal{Y}}\cap I_{\mathcal{V}}$
and so $(I_{\mathcal{Y}}\cap I_{\mathcal{V}}:f)=(I_{\mathcal{Y}}:f)$. Moreover
the latter equals $I_{\mathcal{Y}}$, since $\mathcal{Y}$ is flat over $S$, so
that multiplication by the non-zero-divisor $f$ remains injective after tensor
product with $\mathcal{O}_{\mathcal{Y}}$, that is $A/I_{\mathcal{Y}}$. Thus
$\mathcal{J}$ is locally

$\displaystyle(I_{\mathcal{Y}}\cap I_{\mathcal{V}}:f)/(I_{\mathcal{Y}}\cap
I_{\mathcal{V}})$ $\displaystyle=I_{\mathcal{Y}}/(I_{\mathcal{Y}}\cap
I_{\mathcal{V}})$
$\displaystyle\cong(I_{\mathcal{Y}}+I_{\mathcal{V}})/I_{\mathcal{V}}=I_{\mathcal{Y}_{E}}/I_{\mathcal{V}}$

where we write $\mathcal{Y}_{E}$ for the restriction $\mathcal{Y}\cap(X\times
E)=\mathcal{Y}\cap\mathcal{V}$. This shows

$\mathcal{T}\\!\mathit{or}_{1}^{X\times
S}(\mathcal{O}_{\mathcal{Y}\cup\mathcal{V}},\mathcal{O}_{X\times
E})\cong\mathcal{I}_{\mathcal{Y}_{E}/\mathcal{V}}(-\mathit{pr}_{2}^{*}E)$

and (4.5) gives the short exact sequence

$0\to\mathcal{I}_{\mathcal{Y}_{E}/\mathcal{V}}(-\mathit{pr}_{2}^{*}E)\to\mathcal{I}_{\mathcal{Y}\cup\mathcal{V}}\vline_{E}\to\mathcal{I}_{\mathcal{V}}\to
0$

on $X\times E$. Finally restrict to the fiber over a point $\xi\in E$: since
$\mathcal{Y}_{E}$ and $\mathcal{V}$ are both $E$-flat this yields the short
exact sequence in the statement. ∎

Lemma 4.9 does not guarantee that the short exact sequence obtained is
nonsplit. Showing this in the case at hand requires some work. Our strategy is
to exhibit a certain quotient sheaf
$\mathcal{I}_{\mathcal{Y}\cup\mathcal{V}}\otimes
k(\xi)\twoheadrightarrow\mathcal{Q}$ and check that the split extension
$\mathcal{I}_{\mathcal{V}_{\xi}}\oplus\mathcal{I}_{\mathcal{Y}_{\xi}/\mathcal{V}_{\xi}}$
admits no surjection onto $\mathcal{Q}$. In fact
$\mathcal{Q}=\mathcal{O}_{V}(-2)$ will work:

######  Lemma 4.10.

Let $V\subset\mathbb{P}^{3}$ be a plane and $P\in V$ a point. Then there is no
surjection from $\mathcal{O}_{\mathbb{P}^{3}}(-1)\oplus\mathcal{I}_{P/V}(-2)$
onto $\mathcal{O}_{V}(-2)$.

###### Proof.

Just note that
$\operatorname{Hom}(\mathcal{I}_{P/V}(-2),\mathcal{O}_{V}(-2))=k$ is generated
by the (nonsurjective) inclusion, whereas
$\operatorname{Hom}(\mathcal{O}_{\mathbb{P}^{3}}(-1),\mathcal{O}_{V}(-2))=0$.
∎

We will produce the required quotient sheaf by the following construction,
which depends on the choice of a tangent direction at $\xi$ in $S$:

######  Lemma 4.11.

With notation as in Lemma 4.9, let $T=\operatorname{Spec}k[t]/(t^{2})$ and let
$T\subset S$ be a closed embedding such that $T\cap E$ is the reduced point
$\\{\xi\\}$. Let $Y=\mathcal{Y}_{\xi}$ and $V=\mathcal{V}_{\xi}$.

Define a subscheme $Y^{\prime}\subset Y$ by the ideal

$(\mathcal{I}:t)/(t)\subset\mathcal{O}_{V}$

where $\mathcal{I}\subset\mathcal{O}_{V\times T}$ is the ideal of
$\mathcal{Y}\cap(V\times T)$. Then there is a surjection

$\mathcal{I}_{\mathcal{Y}\cup\mathcal{V}}\otimes
k(\xi)\twoheadrightarrow\mathcal{I}_{Y^{\prime}/V}.$

######  Remark 4.12.

Since $t^{2}=0$ we trivially have $t\in(\mathcal{I}:t)$. Since also
$\mathcal{I}\subset(\mathcal{I}:t)$ we furthermore have $Y^{\prime}\subset Y$.
If we extend $T$ to an actual one parameter family of objects $Y_{t}$, we may
think of $Y^{\prime}$ as the limit of $Y_{t}\cap V$ as $t\to 0$, in other
words it is the part of $Y$ that remains in $V$ as we deform along our chosen
direction.

###### Proof.

Let $\mathcal{Y}_{T}\subset\mathbb{P}^{3}\times T$ denote the restriction of
$\mathcal{Y}$ to $T$. We _claim_ that $\mathcal{I}_{Y^{\prime}/V}$ is
isomorphic to the relative ideal of $\mathcal{Y}_{T}\cup(V\times\\{\xi\\})$ in
$\mathcal{Y}_{T}\cup(V\times T)$. Assuming this, there are surjections

$\mathcal{I}_{\mathcal{Y}\cup\mathcal{V}}\otimes\mathcal{O}_{T}\twoheadrightarrow\mathcal{I}_{\mathcal{Y}_{T}\cup(V\times\\{\xi\\})}\twoheadrightarrow\mathcal{I}_{Y^{\prime}/V}$

(the middle term is the ideal of
$\left.{(\mathcal{Y}\cup\mathcal{V})}\right|_{T}=\mathcal{Y}_{T}\cup(V\times\\{\xi\\})$
as a subscheme of $\mathbb{P}^{3}\times T$). Restriction to the fiber over
$\xi$ gives the surjection in the statement.

To prove the claim, we first observe that for any two subschemes $A$ and $B$
of some ambient scheme, there is an isomorphism

$\mathcal{I}_{(A\cap B)/A}\cong\mathcal{I}_{B/(A\cup B)}$

between the relative ideal sheaves; this is the identity $(I+J)/I\cong
J/(I\cap J)$ between quotients of ideals. Apply this to

$A=V\times T,\quad B=\mathcal{Y}_{T}\cup(V\times\\{\xi\\})$

so that

$\displaystyle A\cup B$ $\displaystyle=\mathcal{Y}_{T}\cup(V\times T)$
$\displaystyle A\cap B$
$\displaystyle=(\mathcal{Y}_{T}\cup(V\times\\{\xi\\}))\cap(V\times
T)=(\mathcal{Y}\cap(V\times T))\cup(V\times\\{\xi\\}).$

The _claim_ as stated thus says that $\mathcal{I}_{Y^{\prime}/V}$ is
isomorphic to $\mathcal{I}_{B/A\cup B}$, and we are free to replace the latter
by $\mathcal{I}_{(A\cap B)/A}$.

Next let $\operatorname{Spec}R$ be an affine open subset in $V$ and $I\subset
R[t]/(t^{2})$ the ideal defining $\mathcal{Y}\cap(V\times T)$ there. Locally
the ideal $\mathcal{I}_{(A\cap B)/A}$ is then $I\cap(t)\subset R[t]/(t^{2})$.
Now multiplication with $t$ is an isomorphism of $R[t]/(t^{2})$-modules

$(I:t)/(t)\cong I\cap(t).$

The left hand side is precisely $\mathcal{I}_{Y^{\prime}/V}$ in the open
subset $\operatorname{Spec}R$. ∎

###### Proof of Theorem 4.8 (i).

By Lemma 4.9 it suffices to show that the fiber of
$\mathcal{I}_{\mathcal{Y}\cup\mathcal{V}}\otimes k(\xi)$ over $\xi\in E$ is
isomorphic to $\mathcal{F}_{P,V}$. Moreover, for such $\xi$, the same Lemma
yields a short exact sequence

$0\to\mathcal{I}_{P/V}(-2)\to\mathcal{I}_{\mathcal{Y}\cup\mathcal{V}}\otimes
k(\xi)\to\mathcal{O}_{\mathbb{P}^{3}}(-1)\to 0$

and since $\mathcal{F}_{P,V}$ is the unique such nonsplit extension it is
enough to show that the above extension is nonsplit. In view of Lemma 4.10
this follows once we can show the existence of a surjection

$\mathcal{I}_{\mathcal{Z}\cup\mathcal{W}}\otimes
k(\xi)\twoheadrightarrow\mathcal{O}_{V}(-2).$

For this it suffices, in the notation of Lemma 4.11, to choose
$T\subset\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$ such that the subscheme
$Y^{\prime}\subset V$ is a conic.

_Nondegenerate case._ First assume $Y=\mathcal{Y}_{\xi}$ is a disjoint union
$Y=C\cup\\{P\\}$ of a conic $C\subset V$ and a point $P\in V$. Consider the
one parameter family $Y_{t}=C\cup\\{P_{t}\\}$ in which the conic part $C$ is
fixed while the point $P_{t}$ travels along a line intersecting $V$ in the
point $P$. In suitable affine coordinates we may take $V$ to be the plane
$z=0$ in $\mathbb{A}^{3}=\operatorname{Spec}k[x,y,z]$, the point $P$ to be the
origin and $C$ to be given by some quadric $q=q(x,y)$ not vanishing at $P$.
Let the one parameter family over $\operatorname{Spec}k[t]$ consist of the
union of $C$ with the point $P_{t}=(0,0,t)$. This is given by the ideal

$(q,z)\cap(x,y,z-t)=(xq,yq,(z-t)q,xz,yz,(z-t)z).$

Now restrict to $T=\operatorname{Spec}k[t]/(t^{2})$ and intersect the family
with $V\times T$. The resulting subscheme is defined by the ideal

$I=(xq,yq,(z-t)q,xz,yz,(z-t)z)+(z)=(xq,yq,tq,z)$

and $(I:t)/(t)=(q,z)$ which defines $C\subset V$. Thus $Y^{\prime}=C$ and we
are done.

_Embedded point with nonsingular support._ Suppose $Y$ is a conic $C\subset V$
with an embedded point supported at a point $P$ in which $C$ is nonsingular,
where the normal direction corresponding to the embedded point is along $V$.
Then take the one parameter family in which $C$ and the supporting point $P$
is fixed and the embedded structure varies in the $\mathbb{P}^{1}$ of normal
directions. In suitable affine coordinates we may take $V$ to be the plane
$z=0$ in $\mathbb{A}^{3}=\operatorname{Spec}k[x,y,z]$, $P$ to be the origin
and $C$ given by a quadric $q=q(x,y)$ vanishing at $P$ and with, say, linear
term $y$. Take the one parameter family of $C$ with an embedded point given by

$(q,z)\cap(x,y^{2},z-ty)=(xq,yq,zq,z-tq)$

(the equality requires some computation). After intersection with $V\times T$
this gives

$I=(xq,yq,tq,z)$

and $(I:t)/(t)=(q,z)$. This is $C$.

_Embedded point at a singularity._ Let $C\subset V$ be the union of two
distinct lines intersecting in $P$ and consider a planar embedded point at
$P$. Despite the singularity, there is still a $\mathbb{P}^{1}$ of embedded
points at $P$. We take this to be our one parameter family, i.e. we deform the
embedded point structure away from the planar one.

In local coordinates we take $P$ to be the origin in $\mathbb{A}^{3}$ and $C$
to be the union of the $x$\- and $y$-axes in the $xy$-plane $V$. Then

$(xy,z)(x,y,z)+(z-txy)=(xy^{2},x^{2}y,z-txy)$

is our one parameter family of embedded points at the origin, with $t=0$
corresponding to the planar embedded point. The intersection with $V\times T$
is given by

$I=(xy^{2},x^{2}y,z,txy)$

and $(I:t)/(t)=(xy,z)$. This is $C$.

_Embedded point in a double line._ Let $C\subset V$ be a planar double line
together with a planar embedded point at $P\in C$ and take the one parameter
family of embedded points in $P$.

In local coordinates we take $P$ to be the origin in $\mathbb{A}^{3}$ and $C$
to be $V(z,y^{2})$. Then

$(z,y^{2})(x,y,z)+(z-ty^{2})=(y^{3},xy^{2},z-ty^{2})$

is our one parameter family of embedded points at the origin, with $t=0$
corresponding to the planar embedded point. The intersection with $V\times T$
is given by

$I=(xy^{2},y^{3},z,ty^{2})$

and $(I:t)/(t)=(z,y^{2})$. This is $C$. ∎

###### Proof of Theorem 4.8 (ii).

The morphism
$\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})\to\mathcal{M}^{\mathrm{II}}$ is clearly
an isomorphism away from $E$, and it sends $\xi\in E$ (lying over $(P,V)\in
I$) to $\mathcal{F}_{P,V}$, which determines and is uniquely determined by
$(P,V)$. Moreover $\mathcal{M}^{\mathrm{II}}$ is smooth at these points by
Proposition 3.7. The claim follows from uniqueness of normal contractions. ∎

###  4.3 Moduli in chamber III

In this section we show that the moduli space $\mathcal{M}^{\mathrm{III}}$ is
a contraction of $\mathcal{S}$. The argument parallels that for
$\mathcal{M}^{\mathrm{II}}$ closely.

Let $F\subset\mathcal{S}$ be as in Notation 1.2. Thus an element
$Y\in\mathcal{S}$ is either a pair of intersecting lines with a spatial
embedded point at the intersection, or as degenerate cases, a planar double
line with a spatial embedded point. It is in a natural way a
$\mathbb{P}^{2}$-bundle over the incidence variety
$I\subset\mathbb{P}^{3}\times\check{\mathbb{P}}^{3}$ via the map

$F\to I$

that sends $Y$ to the pair $(P,V)$ consisting of the support $P\in Y$ of the
embedded point and the plane $V$ containing $Y\setminus\\{P\\}$. In parallel
with Proposition 4.3 one may show that $\mathcal{O}_{\mathcal{S}}(F)$
restricts to $\mathcal{O}_{\mathbb{P}^{2}}(-1)$ in the fibers of $F/I$ and so
there is a contraction

$\psi\colon\mathcal{S}\to\mathcal{S}^{\prime}$ (4.6)

to a smooth algebraic space $\mathcal{S}^{\prime}$, such that $\psi$ is an
isomorphism away from $F$ and restricts to the $\mathbb{P}^{2}$-bundle $F\to
I$. However, in this case we can be much more concrete thanks to the work of
Chen–Coskun–Nollet [8], where birational models for $\mathcal{S}$ are studied
in detail (and in greater generality: moduli spaces for pairs of codimension
two linear subspaces of projective spaces in arbitrary dimension). The
following proposition is [8, Theorem 1.6 (4)]; we sketch a simple and slightly
different argument here.

######  Proposition 4.13.

There is a contraction as in (4.6) where $\mathcal{S}^{\prime}$ is the
Grassmannian $G(2,6)$ of lines in $\mathbb{P}^{5}$.

###### Proof.

First consider an arbitrary quadric $Q\subset\mathbb{P}^{n}$. Any finite
subscheme in $Q$ of length $2$, reduced or not, determines a line in
$\mathbb{P}^{n}$. This defines a morphism

$\mathrm{Hilb}^{2}(Q)\to G(2,n+1).$ (4.7)

It is clearly an isomorphism away from the locus in $G(2,n+1)$ consisting of
lines contained in $Q$. On the other hand, over every element of $G(2,n+1)$
defining a line contained in $Q$, the fiber is the $\mathbb{P}^{2}$ consisting
of length two subschemes of that line.

Apply the above observation to the (Plücker) quadric $Q=G(2,4)$ in
$\mathbb{P}^{5}$, so that $\mathcal{S}\cong\mathrm{Hilb}^{2}(Q)$ (see Section
2.1). For every plane $V\subset\mathbb{P}^{3}$ and every point $P\in V$, the
pencil of lines in $V$ through $P$ defines a line in $Q=G(2,4)$ and in fact
every line is of this form. The fiber of (4.7) above such an element of
$G(2,5)$ consists of all pairs of lines in $V$ intersecting at $P$. It follows
that (4.7) is the required contraction $\mathcal{S}\to\mathcal{S}^{\prime}$. ∎

######  Remark 4.14.

Chen–Coskun–Nollet furthermore shows that (4.6) is a $K$-negative extremal
contraction in the sense of Mori theory. In fact, $\mathcal{S}$ is Fano and
its Mori cone is spanned by two rays. Either ray is thus contractible; one
contraction is (4.6) and the other is the natural map to the symmetric square
of the Grassmannian of lines in $\mathbb{P}^{3}$. This statement is extracted
from Theorem 1.3, Lemma 3.2 and Proposition 3.3 in _loc. cit._ Inspired by
this work we return to the Mori cone of the conics-with-a-point component
$\mathcal{C}$ in Section 5.

We proceed as for chamber II by modifying the universal family of pairs of
lines in order to identify the moduli space $\mathcal{M}^{\mathrm{III}}$ with
the contracted space $\mathcal{S}^{\prime}$. Let

$\mathcal{Y}\subset\mathbb{P}^{3}\times\mathcal{S}$

be the restriction of the universal family over
$\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$ to the component $\mathcal{S}$.
Moreover, there is a flat family over the incidence variety
$I\subset\mathbb{P}^{3}\times\check{\mathbb{P}}^{3}$ whose fiber over $(P,V)$
is the plane $V$ with an embedded point at $P$. Pull this back to $F$ to
define a family

$\mathcal{W}\subset\mathbb{P}^{3}\times F.$

We argue as in Section 4.2 but with the family of planes $\mathcal{V}$
replaced by the family of planes with an embedded point $\mathcal{W}$.

######  Theorem 4.15.

Let $\mathcal{Y}$ and $\mathcal{W}$ be as above.

  * (i)

$\mathcal{I}_{\mathcal{Y}\cup\mathcal{W}}$ is flat as a coherent sheaf over
$\mathcal{S}$. Its fibers $\mathcal{I}_{\mathcal{Y}\cup\mathcal{W}}\otimes
k(\xi)$ over $\xi\in\mathcal{S}$ are stable objects for stability conditions
in chamber III.

  * (ii)

The morphism $\mathcal{S}\to\mathcal{M}^{\mathrm{III}}$ determined by
$\mathcal{I}_{\mathcal{Y}\cup\mathcal{W}}$ induces an isomorphism
$\mathcal{S}^{\prime}\cong\mathcal{M}^{\mathrm{III}}$.

For $\xi\in F$ lying over $(P,V)$ we have
$\mathcal{I}_{\mathcal{Y}_{\xi}/\mathcal{W}_{\xi}}\cong\mathcal{O}_{V}(-2)$
and $\mathcal{I}_{\mathcal{W}_{\xi}}\cong\mathcal{I}_{P}(-1)$. Thus Lemma 4.9
yields a short exact sequence

$0\to\mathcal{O}_{V}(-2)\to\mathcal{I}_{\mathcal{Y}\cup\mathcal{W}}\otimes
k(\xi)\to\mathcal{I}_{P}(-1)\to 0$

and we show that it is nonsplit by exhibiting a certain quotient sheaf of
$\mathcal{I}_{\mathcal{Y}\cup\mathcal{W}}\otimes k(\xi)$. This time we use
$\mathcal{I}_{Q/V}(-1)$ where $Q\in V$ is a point distinct from $P$.

######  Lemma 4.16.

Let $V\subset\mathbb{P}^{3}$ be a plane and $P,Q\in V$ two distinct points.
There is no surjection from $\mathcal{O}_{V}(-2)\oplus\mathcal{I}_{P}(-1)$ to
$\mathcal{I}_{Q/V}(-1)$.

###### Proof.

Every nonzero homomorphism

$\mathcal{O}_{V}(-2)\to\mathcal{O}_{V}(-1)$

has image of the form $\mathcal{I}_{L/V}(-1)$ where $L\subset V$ is a line,
whereas every nonzero homomorphism

$\mathcal{I}_{P}(-1)\to\mathcal{O}_{V}(-1)$

has image $\mathcal{I}_{P/V}(-1)$. Thus any nonzero homomorphism from the
direct sum of these two sheaves has image $\mathcal{I}(-1)$ where
$\mathcal{I}\subset\mathcal{O}_{V}$ is one of $\mathcal{I}_{L/V}$,
$\mathcal{I}_{P/V}$ or their sum

$\mathcal{I}_{L/V}+\mathcal{I}_{P/V}=\begin{cases}\mathcal{I}_{P/V}&\text{if
$P\in L$}\\\ \mathcal{O}_{V}&\text{otherwise.}\end{cases}$

Thus the image is never $\mathcal{I}_{Q/V}(-1)$ for $Q\neq P$. ∎

###### Proof of Theorem 4.15.

The proof for Theorem 4.8 carries over; we only need to detail the
construction of quotient sheaves via one parameter families. As before we
write down families over $\mathbb{A}^{1}=\operatorname{Spec}k[t]$ and then
restrict to $T=\operatorname{Spec}k[t]/(t^{2})$. We then apply Lemma 4.11,
with $\mathcal{W}$ in the role of the family denoted $\mathcal{V}$ in the
Lemma. The outcome of Lemma 4.11 will be a quotient sheaf of the form
$\mathcal{I}_{Y^{\prime}/W}$, where $W=\mathcal{W}_{\xi}$ is a plane $V$ with
an embedded point at $P$. We end by intersecting with $V$ to produce a further
quotient of the form $\mathcal{I}_{Y^{\prime}\cap V/V}$. We shall choose one
parameter families such that the latter is isomorphic to
$\mathcal{I}_{Q/V}(-1)$ with $Q\neq P$.

_Distinct lines._ Let $C=L\cup L_{0}$ be a pair of distinct lines inside $V$
intersecting at $P$. Choose another plane $V^{\prime}$ containing $L_{0}$ and
a point $Q\in L_{0}$ distinct from $P$. The pencil of lines $L_{t}\subset
V^{\prime}$ through $Q$ yields a one parameter family

$Z_{t}=L\cup L_{t}$

of disjoint pairs of lines for $t\neq 0$, with flat limit $Z_{0}\subset W$
being $C$ with a spatial embedded point at $P$.

In suitable affine coordinates $\mathbb{A}^{3}$ let $V$ be $V(z)$, let $P$ be
the origin and let $C=V(z,xy)$. Then $W=V(xz,yz,z^{2})$. Furthermore let
$Q=(0,1,0)$ and $L_{t}=V(x,z-t(y-1))$. This leads to the family $Z$ defined by
the ideal

$(y,z)\cap(x,z-t(y-1))=(xy,xz,(z-t(y-1))y,(z-t(y-1))z)$

and the intersection with $W\times T$ is given by

$I=(xz,yz,z^{2},xy,ty(y-1),tz)$

Thus $(I:t)/(t)=(z,xy,y(y-1))$, which defines the union of the $x$-axis and
the point $Q$. This is $Y^{\prime}\subset W$ and thus
$\mathcal{I}_{Y^{\prime}\cap V/V}=\mathcal{I}_{Y^{\prime}/V}$ is isomorphic to
$\mathcal{I}_{Q/V}(-1)$.

_Double lines._ Let $C\subset V$ be a double line inside the plane $V$ with
$P\in C$. We shall define an explicit one parameter family with central fiber
$Y_{0}\subset W$ being $C$ with a spatial embedded point at $P$.

Geometrically, the family is this: let $L\subset V$ be the supporting line of
$C$. Consider a line $M\subset V$ not through $P$ and let $Q$ be its
intersection point with $L$. Also let $M^{\prime}$ be a line through $P$ and
not contained in $V$. Let $R_{t}$ be a point on $M$ moving towards $Q$ as
$t\to 0$, and let $R^{\prime}_{t}$ be a point on $M^{\prime}$ moving towards
$P$, but much faster than $R_{t}$ moves (quadratic versus linearly). Then let
$L_{t}$ be the line through $R_{t}$ and $R^{\prime}_{t}$ and let $Y_{t}=L\cup
L_{t}$ for $t\neq 0$.

Let $P$ be the origin in suitable affine coordinates $\mathbb{A}^{3}$, let $V$
be the $xy$-plane $V(z)$ and let $C\subset V$ be the double $x$-axis
$V(y^{2},z)$. Thus $Y_{0}$ corresponds to

$(z,y^{2})\cap(x,y,z)^{2}=(xz,yz,z^{2},y^{2}).$

Now let $L=V(y,z)$, let $L_{t}$ be the line through $(1,t,0)$ and
$(0,0,t^{2})$, that is

$L_{t}=V(tx-y,ty+z-t^{2})$

and take $Y_{t}=L\cup L_{t}$ for $t\neq 0$. This yields the family (the
following identity requires a bit of fiddling)

$(y,z)\cap(tx-y,ty+z-t^{2})=((tx-y)y,(tx-y)z,(ty+z-t^{2})z,xz+ty(x-1)).$

Reducing this modulo $t$ gives the original $Y_{0}$. The intersection with
$W\times T$ gives

$I=(xz,yz,z^{2},(tx-y)y,ty(x-1))$

and so $Y^{\prime}\subset W$ is defined by

$\displaystyle(I:t)/(t)$ $\displaystyle=(xz,yz,z^{2},y^{2},y(x-1))$
$\displaystyle=(y,z)\cap(x-1,y^{2},z)\cap(x,y,z^{2}).$

This is the line $L$ with an embedded point at $Q$ (inside $V$) and another
embedded point along the $z$-axis at $P$. Intersecting with $V$ removes the
embedded point at $P$, leaving the line $L$ with an embedded point at $Q$.
Thus $\mathcal{I}_{Y^{\prime}\cap V/V}\cong\mathcal{I}_{Q/V}(-1)$.

This establishes part (i) precisely as in the proof of Theorem 4.8 and part
(ii) then follows by smoothness of $\mathcal{M}^{\mathrm{III}}$ (from
Proposition 3.9) and by uniqueness of normal (here smooth) contractions. ∎

##  5 The Mori cone of $\mathcal{C}$ and extremal contractions

In this final section we shall prove that $\mathcal{C}\to\mathcal{C}^{\prime}$
is the contraction of a $K$-negative extremal ray in the Mori cone. It follows
that the contracted space $\mathcal{C}^{\prime}$ is projective.

To set the stage we recall the basic mechanism of $K$-negative extremal
contractions. Let $X$ be a projective normal variety and $\alpha$ a curve
class (modulo numerical equivalence) which spans an extremal ray in the Mori
cone. If also the ray is $K$-negative, i.e. the intersection number between
$\alpha$ and the canonical divisor $K_{X}$ is negative, then there exists a
unique projective normal variety $Y$ and a birational morphism $f\colon X\to
Y$ which contracts precisely the effective curves in the class $\alpha$.

###  5.1 Statement

We denote elements in $\mathcal{C}$ by the letter $Y$. It is the union of a
(possibly degenerate) conic denoted $C$ and a point denoted $P$. If the point
is embedded, $P\in C$ denotes its support. We also write
$V\subset\mathbb{P}^{3}$ for the unique plane containing $C$.

Define four effective curve classes (modulo numerical equivalence) on
$\mathcal{C}$. Each is described as a family $\\{Y_{t}\\}$, and we use a
subscript $t$ to indicate a parameter on the piece that varies (all choices
are to be made general, e.g. $C$ nonsingular unless stated otherwise, etc.):

  1. $\delta$:

fix a conic $C$ and a point $P\in C$. Let $Y_{t}$ be $C$ with an embedded
point at $P$, varying in the $\mathbb{P}^{1}$ of normal directions to
$C\subset\mathbb{P}^{3}$ at $P$.

  2. $\epsilon$:

fix a plane $V$, a conic $C\subset V$ and a line $L\subset V$. Let $P_{t}$
vary along $L$ and let $Y_{t}=C\cup\\{P_{t}\\}$.

  3. $\zeta$:

fix a plane $V$, a pencil of conics $C_{t}\subset V$ and a point $P\in V$. Let
$Y_{t}=C_{t}\cup\\{P\\}$.

  4. $\eta$:

fix a line $L$ and a point $P\in L$. Let $V_{t}$ be the pencil of planes
containing $L$ and let $C_{t}$ be the planar double structure on $L$ inside
$V_{t}$. Then let $Y_{t}$ be $C_{t}$ with an embedded spatial point at $P$.

In $\epsilon$ there are implicitly two elements with an embedded point, namely
where $L$ intersects $C$. Similarly there is one element in $\zeta$ with an
embedded point, corresponding to the pencil member $C_{t}$ that contains $P$.

######  Theorem 5.1.

The Mori cone of $\mathcal{C}$ is the cone over a solid tetrahedron, with
extremal rays spanned by the four curve classes $\delta,\epsilon,\zeta,\eta$.
Of these, the first three are $K$-negative, whereas $\eta$ is $K$-positive.
The contraction corresponding to $\zeta$ is
$\mathcal{C}\to\mathcal{C}^{\prime}$.

######  Corollary 5.2.

$\mathcal{C}^{\prime}$ is projective.

######  Remark 5.3.

The last claim in the theorem is clear: by contracting $\zeta$ we forget the
conic part of $Y\subset V$, keeping only $V$ and the point $P\in V$. By
uniqueness of (normal) contractions the contracted variety is
$\mathcal{C}^{\prime}$. Also, with reference to Diagram 4.1 (from Section
4.1), the contraction of $\delta$ is the blowing down $b$. The theorem
furthermore reveals a third $K$-negative extremal ray spanned by $\epsilon$.
The corresponding contraction has the effect of forgetting the point part of
$Y\subset V$, keeping only the conic; thus the contracted locus in
$\mathcal{C}$ is the same as for $\zeta$, but the contraction happens in a
“different direction”. We do not know if the contracted space has an
interpretation as a moduli space for Bridgeland stable objects.

###  5.2 The canonical divisor

Use notation as in Diagram 4.1 and Lemma 4.2. We read off that the Picard
group of $\mathcal{C}$ has rank $4$ and is generated by the pullbacks of the
following divisor classes:

$\displaystyle H\subset\mathbb{P}^{3}$ a plane, $\displaystyle
H^{\prime}\subset\check{\mathbb{P}}^{3}$ a plane in the dual space,
$\displaystyle A=c_{1}(\mathcal{O}_{\mathbb{P}(\mathcal{E}^{\vee})}(1)),$
$\displaystyle E^{\prime}\subset\mathcal{C}$ the exceptional divisor for the
blowup $b$.

Moreover numerical and linear equivalence of divisors coincide on
$\mathcal{C}$. Here we only use that the Picard group of a projective bundle
over some variety $X$ is $\mathrm{Pic}(X)\oplus\mathbb{Z}$, with the added
summand generated by $\mathcal{O}(1)$, and the Picard group of a blowup of $X$
is $\mathrm{Pic}(X)\oplus\mathbb{Z}$, with the added summand generated by the
exceptional divisor.

As long as confusion seems unlikely to occur we will continue to use the
symbols $H$, $H^{\prime}$ and $A$ for their pullbacks to $\mathcal{C}$, or to
an intermediate variety such as
$\mathbb{P}^{3}\times\mathrm{Hilb}^{2m+2}(\mathbb{P}^{3})$ in Diagram 4.1.

######  Lemma 5.4.

The canonical divisor class of $\mathcal{C}$ is

$K_{\mathcal{C}}=-4H-8H^{\prime}-6A+E^{\prime}.$

###### Proof.

This is a standard computation. First, for the blowup $b$, with center of
codimension two, we have

$K_{\mathcal{C}}=b^{*}K_{\mathbb{P}^{3}\times\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})}+E^{\prime}$

and for the product

$K_{\mathbb{P}^{3}\times\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})}=\mathit{pr}_{1}^{*}K_{\mathbb{P}^{3}}+\mathit{pr}_{2}^{*}K_{\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})}.$

Now $K_{\mathbb{P}^{3}}=-4H$ and for the projective bundle
$\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})\cong\mathbb{P}(\mathcal{E}^{\vee})$ we
have

$K_{\mathbb{P}(\mathcal{E}^{\vee})}=\pi^{*}K_{\check{\mathbb{P}}^{3}}+c_{1}(\Omega^{1}_{\pi}).$

Again $K_{\check{\mathbb{P}}^{3}}=-4H^{\prime}$ and the short exact sequence

$0\to\Omega^{1}_{\pi}\to\pi^{*}(\mathcal{E}^{\vee})\otimes\mathcal{O}_{\mathbb{P}(\mathcal{E}^{\vee})}(-1)\to\mathcal{O}_{\mathbb{P}(\mathcal{E}^{\vee})}\to
0$

gives

$\displaystyle c_{1}(\Omega^{1}_{\pi})$
$\displaystyle=c_{1}(\pi^{*}(\mathcal{E}^{\vee})\otimes\mathcal{O}_{\mathbb{P}(\mathcal{E}^{\vee})}(-1))$
$\displaystyle=\pi^{*}c_{1}(\mathcal{E}^{\vee})+6c_{1}(\mathcal{O}_{\mathcal{E}^{\vee}}(-1)$
$\displaystyle=-\pi^{*}c_{1}(\mathcal{E})-6A.$

Putting this together, the stated expression for $K_{\mathcal{C}}$ follows
once we have established that $c_{1}(\mathcal{E})=4H^{\prime}$.

Recall that
$\mathcal{E}=\mathit{pr}_{2*}(\left.{\mathit{pr}_{1}^{*}\mathcal{O}_{\mathbb{P}^{3}}(2)}\right|_{I})$.
We compute its first Chern class by brute force: apply
Grothendieck–Riemann–Roch to
$\mathit{pr}_{2}\colon\mathbb{P}^{3}\times\check{\mathbb{P}}^{3}\to\check{\mathbb{P}}^{3}$.
Note that all higher direct images vanish, since
$H^{p}(V,\mathcal{O}_{V}(2))=0$ for all $V\in\check{\mathbb{P}}^{3}$ and
$p>0$. Thus by Grothendieck–Riemann–Roch the class

$c_{1}(\mathit{pr}_{2*}(\left.{\mathit{pr}_{1}^{*}\mathcal{O}_{\mathbb{P}^{3}}(2)}\right|_{I}))$

is the push forward in the sense of the Chow ring of the degree $4$
homogeneous part of

$\mathrm{ch}(\left.{\mathit{pr}_{1}^{*}\mathcal{O}_{\mathbb{P}^{3}}(2)}\right|_{I})\mathit{pr}_{1}^{*}(\mathrm{td}(\mathbb{P}^{3})).$

We have

$\mathrm{td}(\mathbb{P}^{3})=\left(\frac{H}{1-e^{-H}}\right)^{4}=1+2H+\tfrac{11}{6}H^{2}+H^{3}.$
(5.1)

Moreover $I\subset\mathbb{P}^{3}\times\check{\mathbb{P}}^{3}$ is a divisor of
bidegree $(1,1)$, so there is a short exact sequence

$0\to\mathit{pr}_{1}^{*}\mathcal{O}_{\mathbb{P}^{3}}(-1)\otimes\mathit{pr}_{2}^{*}\mathcal{O}_{\check{\mathbb{P}}^{3}}(-1)\to\mathcal{O}_{\mathbb{P}^{3}\times\check{\mathbb{P}}^{3}}\to\mathcal{O}_{I}\to
0$

from which we see (suppressing the explicit pullbacks $\mathit{pr}_{i}^{*}$ of
cycles in the notation)

$\mathrm{ch}(\left.{\mathit{pr}_{1}^{*}\mathcal{O}_{\mathbb{P}^{3}}(2)}\right|_{I}))=\exp(2H)(1-\exp(-H)\exp(-H^{\prime})).$
(5.2)

Now multiply together (5.1) and (5.2) and observe that the
$H^{3}H^{\prime}$-coefficient is $4$. Since the push forward
$\mathit{pr}_{2*}$ of any degree $4$ monomial $H^{k}H^{\prime 4-k}$ equals
$H^{\prime}$ if $k=3$ and $0$ otherwise, this shows that
$c_{1}(\mathcal{E})=4H^{\prime}$. ∎

###  5.3 Basis for $1$-cycles

We will need a few more effective curves, as before written as families
$\\{Y_{t}\\}$:

  1. $\alpha:$

fix a conic $C$ and a line $L$. Let the point $P_{t}$ vary along $L$ and let
$Y_{t}=C\cup\\{P_{t}\\}$.

  2. $\beta:$

fix a quadric surface $Q\subset\mathbb{P}^{3}$, a line $L$ and a point $P$.
Let $V_{t}$ run through the pencil of planes containing $L$ and let
$C_{t}=Q\cap V_{t}$. Then take $Y_{t}=C_{t}\cup\\{P\\}$.

  3. $\gamma:$

fix a plane $V$ and a point $P$. Let $C_{t}\subset V$ run through a pencil of
conics and let $Y_{t}=C_{t}\cup\\{P\\}$.

As before all choices are general, so that in the definition of $\alpha$, the
line $L$ is disjoint from $C$, etc.

######  Lemma 5.5.

The dual basis to $(H,H^{\prime},A,E^{\prime})$ is
$(\alpha,\beta,\gamma,-\delta)$.

###### Proof.

We need to compute all the intersection numbers and verify that we get $0$ or
$1$ as appropriate. Here it is sometimes useful to explicitly write out the
pullbacks to $\mathcal{C}$, e.g. writing $b^{*}(\mathit{pr}_{1}^{*}(H))$
rather than $H$. We view $\alpha,\beta,\gamma,\delta$ not just as equivalence
classes, but as the effective curves defined above. Only the intersection
numbers involving $\beta$ require some real work, and we will save this for
last.

Intersections with $\alpha$: Since $\mathit{pr}_{1*}(b_{*}(\alpha)))$ is the
line $L\subset\mathbb{P}^{3}$ defining $\alpha$ we have
$b^{*}(\mathit{pr}_{1}^{*}(H))\cdot\alpha=H\cdot L=1$. Similarly
$\mathit{pr}_{2*}(b_{*}(\alpha))=0$ shows that the intersections with
$H^{\prime}$ and $A$ vanish. Finally $\alpha$ has no elements with embedded
points, so is disjoint from $E^{\prime}$.

Intersections with $\gamma$: We have $A\cdot\gamma=1$ because $\gamma$ is a
line in a fiber of the projective bundle $\pi$, whereas $A$ restricts to a
hyperplane in every fiber. The remaining intersection numbers vanish as we can
pick disjoint effective representatives.

Intersections with $\delta$: We have $E^{\prime}\cdot\delta=-1$ as $\delta$ is
a fiber of the blowup $b$ and $E^{\prime}$ is the exceptional divisor. The
remaining divisors $H$, $H^{\prime}$ and $A$ are all pullbacks, i.e. of the
form $b^{*}(?)$ and then $b^{*}(?)\cdot\delta=(?)\cdot b_{*}(\delta)=0$.

Intersections with $\beta$: We can choose $\beta$ to be disjoint from $H$ and
$E^{\prime}$. Moreover $\pi_{*}(\mathit{pr}_{1*}(b_{*}(\beta)))$ is the line
$\check{L}\subset\check{\mathbb{P}}^{3}$ dual to the line $L$ defining
$\beta$. This gives
$b^{*}(\mathit{pr}_{1}^{*}(\pi^{*}(H^{\prime})))\cdot\beta=H^{\prime}\cdot\check{L}=1$.

It remains to verify $A\cdot\beta=0$. The definition of $\beta$ can be
understood as follows: choose a general section

$\mathcal{O}_{\mathbb{P}^{3}}\xrightarrow{\sigma}\mathcal{O}_{\mathbb{P}^{3}}(2)$

and apply $\mathit{pr}_{2*}(\left.{\mathit{pr}_{1}^{*}(-)}\right|_{I})$ to
obtain a homomorphism

$\mathcal{O}_{\check{\mathbb{P}}^{3}}\to\mathcal{E}$ (5.3)

whose fiber over $V\in\check{\mathbb{P}}^{3}$ is exactly the restriction of
$\sigma$ to $V$. This is nowhere zero, so (5.3) is a rank $1$ subbundle and it
defines a section

$s_{Q}\colon\check{\mathbb{P}}^{3}\to\mathbb{P}(\mathcal{E}^{\vee})\cong\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})$

with
$s_{Q}^{*}(\mathcal{O}_{\mathbb{P}(\mathcal{E}^{\vee})}(-1))\cong\mathcal{O}_{\check{\mathbb{P}}^{3}}$
or in terms of divisors $s_{Q}^{*}(A)=0$. If we let $Q\subset\mathbb{P}^{3}$
be the quadric defined by $\sigma$ then $s_{Q}(V)=Q\cap V$. Thus
$\mathit{pr}_{2*}(b_{*}(\beta)))=s_{Q*}(\check{L})$ where $\check{L}$ is the
dual to the line $L$ defining $\beta$. This gives

$b^{*}(\mathit{pr}_{2}^{*}(A))\cdot\beta=A\cdot\mathit{pr}_{2*}(b_{*}(\beta))=A\cdot
s_{Q*}(\check{L})=s_{Q}^{*}(A)\cdot\check{L}=0.$

∎

We also define the following three effective divisors, phrased as a condition
on $Y\in\mathcal{C}$:

  1. $D:$

all $Y$ whose conic part $C$ intersects a fixed line $M\subset\mathbb{P}^{3}$.

  2. $D^{\prime}:$

all $Y$ such that the line through $P$ and a fixed point
$P_{0}\in\mathbb{P}^{3}$ intersects the conic part $C$.

  3. $E:$

all planar $Y$ (as before).

Since $D$ is defined by a condition on $C$ only, it is the preimage by
$\mathit{pr}_{2}\circ b$ (see Diagram 4.1) of the similarly defined divisor in
$\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})$. Moreover $D^{\prime}$ and $E$ are the
_strict_ transforms by $b$ of the similarly defined divisors on
$\mathbb{P}^{3}\times\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})$.

We will need to control elements of $D^{\prime}$ with an embedded point.

######  Lemma 5.6.

Fix $P_{0}$ so that $D^{\prime}$ is defined as an effective divisor. Choose a
plane $V$ not containing $P_{0}$, a possibly degenerate conic $C\subset V$ and
a point $P\in C$. Then there is a unique $Y\in D^{\prime}$ with conic part $C$
and an embedded point at $P$. More precisely:

  1. 1.

If $C$ is nonsingular at $P$ then the embedded point structure is uniquely
determined by the normal direction given by the line through $P_{0}$ and $P$.

  2. 2.

If $C$ is a pair of lines intersecting at $P$ or a double line, then the
embedded point is the spatial one, i.e. the scheme theoretic union of $C$ and
the first order infinitesimal neighborhood of $P$ in $\mathbb{P}^{3}$.

###### Proof.

Let $Q$ be the cone over $C$ with vertex $P_{0}$. This is a quadratic cone in
the usual sense when $C$ is nonsingular, otherwise $Q$ is either a pair of
planes or a double plane. A disjoint union $C\cup\\{P^{\prime}\\}$ with
$P^{\prime}\neq P_{0}$ is clearly in $D^{\prime}$ if and only if it is a
subscheme of $Q$.

On the one hand this shows that the subschemes $Y$ listed in (1) and (2) are
indeed in $D^{\prime}$, since they are obtained from $C\cup\\{P^{\prime}\\}$
by letting $P^{\prime}$ approach $P$ along the line joining $P_{0}$ and $P$.

On the other hand it follows that if $Y\in D^{\prime}$ then $Y\subset Q$,
since the latter is a closed condition on $Y$. In case (1) $Q$ is nonsingular
at $P$ and so there is a unique embedded point structure at $P\in C$ which is
contained in $Q$. In case (2) the following explicit computation gives the
result: suppose in local affine coordinates that $C$ is the pair of lines
$V(xy,z)$, the “vertex” $P_{0}$ is on the $z$-axis and $P$ is the origin. Then
$Q$ is the pair of planes $V(xy)$. Any $C$ with an embedded point at $P$ has
ideal of the form

$(xy,z)(x,y,z)+(sxy+tz)$

for $(s:t)\in\mathbb{P}^{1}$. This contains the defining equation $xy$ of $Q$
if and only if $t=0$, which defines the spatial embedded point. The case where
$C$ is double line $V(x^{2},z)$ is similar. ∎

######  Lemma 5.7.

We have

$\displaystyle D$ $\displaystyle=2H^{\prime}+A,$ $\displaystyle D^{\prime}$
$\displaystyle=2H+2H^{\prime}+A-E^{\prime},$ $\displaystyle E$
$\displaystyle=H+H^{\prime}-E^{\prime}.$

###### Proof.

The last equality was essentially established in the proof of Proposition 4.3:
it follows from the observations (1) $E$ is the strict transform of $b(E)$,
and (2) the latter is the pullback of the incidence variety
$I\subset\mathbb{P}^{3}\times\check{\mathbb{P}}^{3}$ which is linearly
equivalent to $H+H^{\prime}$.

The remaining two identities are verified by computing the intersection
numbers with the curves in the basis from Lemma 5.5. All curves and divisors
involved are concretely defined and it is easy to find and count the
intersections directly. Some care is needed to rule out intersection
multiplicities, and we often find it most efficient to resort to a computation
in local coordinates. We limit ourselves to writing out only two cases.

The case $D\cdot\beta=2$: As we noted $D$ is really a divisor on
$\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})$ and so we shall write it here as
$b^{*}(\mathit{pr}_{2}^{*}(D))$. Then
$D\subset\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})$ consists of all conics
intersecting a fixed line $M$. We have

$b^{*}(\mathit{pr}_{2}^{*}(D))\cdot\beta=D\cdot\mathit{pr}_{2*}(b_{*}(\beta))$

and $\mathit{pr}_{2*}(b_{*}(\beta))$ is the family of conics $C_{t}=V_{t}\cap
Q$ where $Q$ is a fixed quadric surface and $V_{t}$ runs through the pencil of
planes containing a fixed line $L$. For general choices $M\cap Q$ consists of
two points, and each point spans together with $L$ a plane. This yields
exactly two planes $V_{0}$ and $V_{1}$ in the pencil for which
$C_{0}=V_{0}\cap Q$ and $C_{1}=V_{1}\cap Q$ intersects $M$. It remains to rule
out multiplicities.

In the local coordinates in Remark 4.1 let $M=V(x_{0},x_{1})$. Then the
intersection between $M$ and the plane
$x_{3}=c_{0}x_{0}+c_{1}x_{1}+c_{2}x_{2}$ is the point $(0:0:1:c_{2})$. Now $D$
is the condition that this point is on $C$, i.e. it satisfies equation (4.3);
this gives that $D$ is $s_{22}=0$. On the other hand,
$\mathit{pr}_{2*}(b_{*}(\beta))$ is a one parameter family in which $c_{i}$
and $s_{ij}$ are functions of degree at most $2$ in the parameter. To stay
concrete, let $Q$ be $\sum_{i}x_{i}^{2}=0$ and let $V_{t}$ be $x_{3}=tx_{2}$.
Substitute $x_{3}=tx_{2}$ in the equation for $Q$ to find $C_{t}=Q\cap V_{t}$.
This gives in particular $s_{22}=1+t^{2}$ and so the intersection with $D$ is
indeed two distinct points, each of multiplicity $1$.

The case $D^{\prime}\cdot\delta=1$: This is essentially Lemma 5.6, but to
ascertain there is no intersection multiplicity to account for we argue
differently. $D^{\prime}$ is the strict transform of the divisor
$b(D^{\prime})\subset\mathbb{P}^{3}\times\mathrm{Hilb}^{2m+1}(\mathbb{P}^{3})$,
which contains the center of the blowup. Since $b(D^{\prime})$ is nonsingular
(pick $P_{0}=(0:0:0:1)$ in the definition of $D^{\prime}$, then in the local
coordinates of Remark 4.1 it is simply given by the equation (4.3)) we have
$D^{\prime}=b^{*}(b(D^{\prime}))-E^{\prime}$. Thus

$D^{\prime}\cdot\delta=b(D^{\prime})\cdot
b_{*}(\delta)-E^{\prime}\cdot\delta=0-(-1)$

and we are done.

The remaining cases are either similar to these or easier. ∎

###  5.4 Nef and Mori cones

It is clear that $H$, $H^{\prime}$ and $D$ are base point free, hence nef. For
instance, consider $D$: given $Y\in\mathcal{C}$, choose a line
$M\subset\mathbb{P}^{3}$ disjoint from $Y\subset\mathbb{P}^{3}$. This defines
an effective representative for $D$ not containing $Y$.

######  Lemma 5.8.

The divisor $D^{\prime}+H^{\prime}$ is nef.

###### Proof.

We begin by narrowing down the base locus of $D^{\prime}$. First consider an
element $Y\in\mathcal{C}$ without embedded point, that is a disjoint union
$Y=C\cup\\{P\\}$. Then choose $P_{0}$ such that the line through $P_{0}$ and
$P$ is disjoint from $C$. This defines a representative for $D^{\prime}$ not
containing $Y$, so $Y$ is not in the base locus.

Next let $Y$ be a conic $C$ with an embedded point at a point $P\in C$ where
$C$ is nonsingular. The tangent to $C$ at $P$ together with the normal
direction given by the embedded point determines a plane. Pick $P_{0}$ such
that the line through $P$ and $P_{0}$ defines a normal direction to $C$ which
is distinct from that defined by the embedded point. This determines a
representative for $D^{\prime}$ which by Lemma 5.6(i) does not contain $Y$, so
$Y$ is not in the base locus.

The remaining possibility is that $Y$ is either a pair of intersecting lines
with an embedded point at the singularity, or a double line with an embedded
point. Pick a representative for $D^{\prime}$ by choosing $P_{0}$ outside the
plane containing the degenerate conic. If the embedded point is not spatial,
then Lemma 5.6(ii) shows that $Y$ is not in $D^{\prime}$. So $Y$ is not in the
base locus unless the embedded point is spatial.

Thus let $B\subset\mathcal{C}$ be the locus of intersecting lines with a
spatial embedded point at the origin, together with double lines with a
spatial embedded point. By the above $B$ contains the base locus of
$D^{\prime}$, so if $T\subset\mathcal{C}$ is an irreducible curve not
contained in $B$ then

$(D^{\prime}+H^{\prime})\cdot T=D^{\prime}\cdot T+H^{\prime}\cdot T\geq 0$

as both terms are nonnegative. If on the other hand $T\subset B$ we observe
that $T\cdot E=0$: in fact $B$ and $E$ are disjoint, since every element in
$B$ has a spatial embedded point, whereas all elements in $E$ are planar. By
the relations in Lemma 5.7

$D^{\prime}+H^{\prime}=H+D+E$

and so, using that $H$ and $D$ are nef,

$(D^{\prime}+H^{\prime})\cdot T=(H+D+E)\cdot T=\underbrace{H\cdot T+D\cdot
T}_{\geq 0}+\underbrace{E\cdot T}_{0}\geq 0.$

∎

######  Lemma 5.9.

The dual basis to $(H,H^{\prime},D,D^{\prime}+H^{\prime})$ is
$(\epsilon,\eta,\zeta,\delta)$.

###### Proof.

Lemma 5.5 together with the relations in Lemma 5.7 implies that
$(D^{\prime}+H^{\prime})\cdot\delta=1$ and the other tree intersection numbers
with $\delta$ vanish.

Of the remaining intersection numbers only those involving $\eta$ requires
some care and we shall write out only those.

A representative for $\eta$ is obtained by fixing a line $L$ and a point $P\in
L$ and letting the plane $V_{t}$ vary in the pencil of planes containing