# Virtual classes for the working mathematician

L.Battistella, F.Carocci, C.Manolache

###### Abstract.

This note is intended to be a friendly introduction to virtual classes. We
review virtual classes and we give a number of properties and applications. We
also include a new virtual push-forward theorem and many computations of
virtual classes in simple examples.

###### Contents

  1. 1 Introduction
  2. 2 Motivation
  3. 3 Normal cones
  4. 4 Virtual pull-backs
  5. 5 Gluing obstruction bundles
  6. 6 Deformations of obstructions and cones
  7. 7 Properties of virtual classes and computational methods
  8. 8 Applications to moduli spaces of stable maps

##  1\. Introduction

Virtual classes have been introduced by Li–Tian [LT98] and Behrend–Fantechi
[BF97] and are a key tool in enumerative geometry. However, virtual classes
are usually considered to be difficult, possibly because there are very few
explicit computations. The goal of this note is to show that virtual classes
of well understood schemes are usually computable. The main reason for the
lack of computations is not the definition of virtual classes – which may
indeed look unsuited for computations – but the poor understanding of spaces
on which we are usually interested to compute virtual classes. The main
examples of spaces with virtual classes are moduli spaces of curves on a given
variety such as moduli spaces of stable maps, Hilbert schemes, etc. All these
spaces have the same major drawback: we typically know very little about their
geometry. We expect that if we had a space with known cohomology and equations
of this space inside a smooth ambient space, then the computation of its
virtual class would be possible (see Section 7.3).

We now briefly describe the content of this paper. One of the ingredients in
the construction of virtual classes is the normal cone. In Section 3 we give
examples of explicit computations of normal cones and we list properties of
normal cones and their Segre classes. The second ingredient is a choice of a
perfect obstruction theory. We define obstruction theories first in Section 4
and then more generally in Section 5. In Section 5 an obstruction theory
$E_{X/V}$ on a scheme or DM stack $X$ is given by the following data:

  * •

a global embedding $X\hookrightarrow V$, with $V$ a smooth scheme or DM stack,

  * •

a covering of $V$ by open sets $V_{i}$, and on each $U_{i}=X\cap V_{i}$ a set
of generators for the ideal $I_{U_{i}/V_{i}}$,

  * •

gluing data for the sets of generators.

We compute examples of virtual classes associated to such data. In the end of
Section 5 we review the more general construction in [BF97]. The main
advantage of the construction in [BF97] is that one does not need a global
embedding of $X$ in a smooth scheme/ stack. In most examples in this note, we
will assume such an embedding exists. This is not a very restrictive
assumption for our purposes; for example moduli spaces of stable maps admit
such embeddings [GP99, Appendix A]. In Section 6 we review a variant of the
deformation to the normal cones in [KKP03] and we use this construction to
compute virtual classes in certain examples. In Section 7 we list properties
of virtual classes and we give two computational methods. Section 7 includes a
new virtual push forward theorem for $X\to Y$, with $X$ a component of $Y$
(Theorem 7.2.5). In Section 8 we give examples of moduli spaces of stable
maps. We discuss the hyperplane property for Gromov–Witten invariants in genus
0 and 1. For genus 1 stable maps we suggest a way to split the virtual class
on the components of the moduli space in a simple example.

### Old and new

For normal cones, Gysin maps and their properties the main reference is
Fulton’s Intersection Theory book [Ful98]. We borrow most of our examples from
it. The stacky version of all the above is explained in [BF97]. The definition
of virtual classes which is closest to our approach is the one of Siebert
[Sie04]. We have tried to give an overview of some of the main constructions
and techniques in virtual intersection theory, but no attempt has been made to
give an exhaustive survey of this topic. There are many fundamental results
which we do not treat here. Among these, we mention the virtual localization
theorem [GP99] of Graber and Pandharipande, which is by far the most used
method to compute virtual classes.

While this note is intended to be expository, there are some results with a
certain degree of originality. The virtual push-forward theorem 7.2.5 and some
ideas in Section 7.3 are new. Section 8 also contains some new examples and
observations.

### Acknowledgements

Most of the material presented here was presented at the Summer School in
Enumerative Geometry at SISSA, Trieste, in 2017. A shorter version was part of
a Lecture series within the Workshop on Moduli and Mirror Symmetry held by
C.M. in Alpensia, in 2016. We thank the organisers of these schools for the
invitation and wonderful work environment. We would also like to thank the
participants for their interest– part of the material in this note grew to
answer the questions asked during these workshops. We thank Tom Coates for
comments on the manuscript.

L.B. is supported by a Royal Society 1st Year URF and DHF Research Grant
Scheme. C.M. is supported by an EPSRC funded Royal Society Dorothy Hodgkin
Fellowship. This work was supported by the Engineering and Physical Sciences
Research Council grant EP/L015234/1: the EPSRC Centre for Doctoral Training in
Geometry and Number Theory at the Interface. This note was finished while L.B
and C.M were in residence at the Mathematical Sciences Research Institute in
Berkeley, California, during the Spring 2018 semester and it is supported by
the National Science Foundation under Grant No. DMS-1440140.

##  2\. Motivation

###  2.1. Enumerative geometry

Suppose we want to count a certain type of smooth curves in a smooth space
$X$. The counting problem is usually solved in two steps: first we construct a
proper parameter space for our curves in $X$, and then we do intersection
theory on this space.

There are by now many proper moduli spaces of curves on a given variety.
Grassmannians have been the first moduli spaces used to solve enumerative
problems. Modern examples of moduli spaces of curves on a given variety
include Hilbert schemes and moduli spaces of stable maps. For a more complete
list see [PT14].

Such moduli spaces are often singular, with many different irreducible
components, possibly of different dimensions. Geometric conditions can often
be translated into cohomology classes, but we need them to act on a
fundamental class in order to get numerical invariants. The solution to this
problem is to construct _virtual fundamental classes_ – these mimic the role
of the fundamental cycle in the smooth/pure dimension case.

After having defined intersection theory on these spaces, we see that we
sometimes get wrong answers to enumerative problems. This is because we also
count _degenerate contributions_ from curves which are not smooth, or have
different numerical invariants. These degenerate contributions come with
different flavours: they could lie on the boundary of our moduli space, or
they could form entire "boundary components".

Virtual classes and degenerate contributions are the main concern of these
notes. Let us look at two elementary examples.

###  2.2. Conics tangent to five lines

Let us determine the number of conics tangent to five general lines in the
plane. This is an example in which the moduli space is irreducible and smooth,
but the intersection is non-transverse and it gives us the wrong answer.

A plane conic is given by a quadratic polynomial

$\sum_{0\leq i\leq j\leq 2}a_{i,j}x_{i}x_{j}$

up to scalar multiplication. The parameter space of conics in $\mathbb{P}^{2}$
is thus identified with $\mathbb{P}^{5}$ with homogeneous coordinates
$[a_{00}:a_{10}:...:a_{22}]$.

To express the tangency condition, consider the line $L_{0}=\\{x_{0}=0\\}$. A
conic is tangent to $L_{0}$ if and only if the polynomial
$a_{11}x_{1}^{2}+a_{12}x_{1}x_{2}+a_{22}x_{2}^{2}$ has a double root, i.e.

$\Delta_{L_{0}}=4a_{11}a_{22}-a_{12}^{2}=0.$

This gives a quadric $Q_{0}$ in $\mathbb{P}^{5}$. We might expect that the
number of conics tangent to five lines in general position is given by
intersecting the corresponding five quadrics. If $H$ denotes the hyperplane
class in $A_{*}(\mathbb{P}^{5})$, then the intersection of five general
quadrics is

$(2[H])^{5}=2^{5}=32.$

This is wrong: the correct number of conics tangent to five given lines is 1
(dually, it is the number of conics through five points). The reason why this
count gives the wrong answer is that the intersection $\bigcap_{i=1}^{5}Q_{i}$
is not transverse: it contains the positive-dimensional locus of double lines
$Z\simeq\mathbb{P}^{2*}$. The contribution of $Z$ to the intersection product
$(2[H])^{5}$ can be computed via residual intersections (see [Ful98, Chapter
9]). We compute this contribution in Example 3.4.3.

###  2.3. A line and a plane

Let us now look at a virtual class on a space with components of different
dimensions. The algebraic construction will be done in Section 4. This example
is modelled on a certain moduli space of stable maps (see Section 8).

Consider $X=V(xz,yz)\subset\mathbb{P}^{3}$, which is the union of the the
plane $H=(z=0)$ and the line $L=(x=y=0)$. Note that $X$ is the zero locus of
two non transverse sections of $\mathcal{O}_{\mathbb{P}^{3}}(2)$. If the two
sections were transverse, their intersection would have been a curve of degree
4 in $\mathbb{P}^{3}$. A virtual class on $X$ can be constructed in the
following way. Deforming the equations which define $X$ to $(xz-\epsilon
y^{2},yz-2\epsilon x^{2})$ we get the union $L\cup C_{\epsilon}$, where
$C_{\epsilon}$ is a twisted cubic in $\mathbb{P}^{3}$. Taking the limit for
$\epsilon$ going to zero, we get the union of the class of $L$ and the class
of a degree 3 plane curve. The class of this limit in $A_{*}(X)$ is a good
candidate for the virtual class of $X$. We will denote it by
$[X]^{\mathrm{vir}}$.

However, deformations are usually hard to keep track of and they would make
our virtual class unsuited for computations in most cases of interest. In the
following we will not construct a class on a deformation of $X$. Instead, we
will use Fulton–Macpherson intersection theory to define a class on our
reducible space.

######  Remark 2.3.1.

Let $X=V(xz,yz)\subset\mathbb{P}^{3}$ as above. If $i:X\to\mathbb{P}^{3}$
denotes the embedding, then we can define $i_{*}[X]^{\mathrm{vir}}$ by

$i_{*}[X]^{\mathrm{vir}}:=c_{2}(\mathcal{O}(2)\oplus\mathcal{O}(2))\cap[\mathbb{P}^{3}].$

The push-forward of $[X]^{\mathrm{vir}}$ is easy to compute and if we are
interested in degree zero intersections of $[X]^{\mathrm{vir}}$ with classes
which are pulled back from classes on $\mathbb{P}^{3}$, we can work with
$i_{*}[X]^{\mathrm{vir}}$ instead of $[X]^{\mathrm{vir}}$.

We do not find this satisfactory. One reason is that we may be interested in
separating $[X]^{\mathrm{vir}}$ on the components of $X$. This information is
lost after pushing forward to $\mathbb{P}^{3}$.

##  3\. Normal cones

The main ingredient in Fulton’s intersection theory is the normal cone of a
closed embedding. We review normal cones and their properties and we give many
examples. The main reference for this section is [Ful98], Chapters 4 and 5. We
assume familiarity with Chow groups (see [Ful98], Chapters 1 and 2).

######  Definition 3.0.1.

Let $X\hookrightarrow Y$ be a closed embedding and
$\mathcal{I}=\mathcal{I}_{X/Y}$ the ideal sheaf of $X$ in $Y$. We define the
_normal cone_ of $X$ in $Y$ as

$C_{X/Y}=\underline{\operatorname{Spec}}_{X}\left(\bigoplus_{n\geq
0}\mathcal{I}^{n}/\mathcal{I}^{n+1}\right)$

and the _normal sheaf_ of $X$ in $Y$ as

$N_{X/Y}=\underline{\operatorname{Spec}}_{X}\left(\mathcal{S}ym^{\bullet}(\mathcal{I}/\mathcal{I}^{2})\right)$

######  Example 3.0.2.

If $X\hookrightarrow Y$ is a regular embedding, then $C_{X/Y}=N_{X/Y}$ and it
is a vector bundle. Furthermore, for $X$ and $Y$ smooth, $N_{X/Y}$ fits in the
exact sequence of vector bundles:

$0\to T_{X}\to T_{Y}\rvert_{X}\to N_{X/Y}\to 0.$

######  Example 3.0.3.

Consider $X=V(xz,yz)\subseteq\mathbb{P}^{3}_{[x:y:z:w]}$. In the following we
compute $C_{X/\mathbb{P}^{3}}$. In the affine chart $w\neq 0$ the coordinate
ring of $X$ is $R=\mathbb{C}[x,y,z]/(xz,yz)$. In this chart, the normal cone
is

$C_{X/\mathbb{P}^{3}}\rvert_{U_{w}}=\operatorname{Spec}\left(R[A,B]/(yA-
xB)\right).$

To see this, consider the surjective morphism

$\displaystyle R[A,B]$ $\displaystyle\rightarrow\bigoplus_{n\geq
0}{I^{n}_{X/\mathbb{P}^{3}}/I^{n+1}_{X/\mathbb{P}^{3}}}$ $\displaystyle A$
$\displaystyle\rightarrow xz$ $\displaystyle B$ $\displaystyle\rightarrow yz.$

Then, $C_{X/\mathbb{P}^{3}}\rvert_{U_{w}}\subset
X\rvert_{U_{w}}\times\mathbb{A}^{2}$ is given by the kernel of this map, which
is generated by $(yA-xB)$. Local descriptions of $C_{X/\mathbb{P}^{3}}$ glue
to give an embedding

$C_{X/\mathbb{P}^{3}}\hookrightarrow\mathcal{O}_{X}(2)\oplus\mathcal{O}_{X}(2).$

The cone $C_{X/\mathbb{P}^{3}}\to X$ has two components as follows.

  * •

Over the line L, the cone is defined by the ring $R[A,B]$. This shows that
over the $z$-axis, $C_{X/\mathbb{P}^{3}}\to X$ is isomorphic to the rank 2
vector bundle $\mathcal{O}_{L}(2)\oplus\mathcal{O}_{L}(2)$.

  * •

Over the plane $H$ we have two different situations: if at least one between
$x$ and $y$ is non zero, say $y$, then the defining ring of the cone is
$R[B]$, i.e. $C_{X/\mathbb{P}^{3}}\to X$ is a line bundle; over the origin the
fibre has rank 2.

We have that $C_{L/\mathbb{P}^{3}}=\mathcal{O}_{L}(1)\oplus\mathcal{O}_{L}(1)$
and the natural map $C_{L/\mathbb{P}^{3}}\to C_{X/\mathbb{P}^{3}}$ collapses
the fiber over the origin $z=0$.This shows that

$C_{X/\mathbb{P}^{3}}\neq C_{L/\mathbb{P}^{3}}\cup C_{H/\mathbb{P}^{3}}.$

Let us now list some general properties of normal cones.

###  3.1. Components of the cone

See [Ful98, §4.1].

  1. (1)

If $Y$ has irreducible components $Y_{1},\dots,Y_{N}$, then

$C_{X/Y}=\bigcup_{i}^{N}C_{X_{i}/Y_{i}},\qquad X_{i}=Y_{i}\times_{Y}X.$

  2. (2)

The number of components of $C_{X/Y}$ is always bigger or equal than the
number of components of $X$, and it may be strictly bigger.

######  Example 3.1.1.

Consider the inclusion of the singular point in the nodal cubic
$X=pt\hookrightarrow Y=V(y^{2}+x^{2}(x-1))$. Then $X$ has only one component
and one can easily compute that $C_{X/Y}=V(y^{2}-x^{2})$ which has two
components.

######  Example 3.1.2.

Let us compute the normal cone of $X=V(x^{2},xy)\subset\mathbb{A}^{2}$. Note
that $X$ is the $y$-axis with an embedded point in the $x$ direction at the
origin.

There is a graded sujective morphism

$\displaystyle\mathbb{C}[x,y]/(x^{2},xy)[A,B]$
$\displaystyle\to\bigoplus_{n\geq
0}{I^{n}_{X/\mathbb{A}^{2}}/I^{n+1}_{X/\mathbb{A}^{2}}}$ $\displaystyle A$
$\displaystyle\to x^{2}$ $\displaystyle B$ $\displaystyle\to xy.$

To understand the ideal of $C_{X/\mathbb{A}^{2}}\subseteq
X\times\mathbb{A}^{2}$ we need to compute $J$ – the kernel of this morphism.
We see that $J$ is generated by $yA-xB$ and thus

$C_{X/\mathbb{A}^{2}}=\operatorname{Spec}\left(\frac{\mathbb{C}[x,y]}{(x^{2},xy)}[A,B]/(yA-
xB)\right).$

Since the only relation we have is linear in $A$ and $B$, the normal cone
coincides with the normal sheaf.

Let us now look at the components of $C_{X/\mathbb{A}^{2}}$. Over the generic
point of the $y$-axis, the fibre of $C_{X/\mathbb{A}^{2}}\to X$ is
$\mathbb{A}^{1}$, so $C_{X/\mathbb{A}^{2}}$ restricted to the punctured line
is a line bundle $L$. The fibre over the origin is isomorphic to
$\mathbb{A}^{2}$. In conclusion, $C_{X/\mathbb{A}^{2}}$ has two irreducible
components, one supported on the origin and another one which is the closure
of $L$ in $X\times\mathbb{A}^{2}$.

###  3.2. Pull-backs

Given a cartesian diagram of schemes

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y}$

there is a closed embedding $C_{X^{\prime}/Y^{\prime}}\hookrightarrow
f^{*}C_{X/Y}.$

###  3.3. Deformation to the normal cone

(see [Ful98, Chapter 5]) Let $X$ be a closed sub-scheme of $Y$ and let
$C_{X/Y}$ be the normal cone of the embedding. There exists a flat family
$\pi\colon M^{0}_{X/Y}\to\mathbb{P}^{1}$ with generic fiber $Y$ and special
fiber $C_{X/Y}$ together with a closed embedding

$X\times\mathbb{P}^{1}\hookrightarrow M^{0}_{X/Y}$

commuting with the maps to $\mathbb{P}^{1}$ such that

  * •

over $\mathbb{P}^{1}\backslash\\{0\\}$ we get the given embedding of $X$ in
$Y$, and

  * •

the fiber over $0$ is the embedding of $X$ into its normal cone as the zero
section.

To construct this deformation, we first define

$M_{X/Y}=\operatorname{Bl}_{X\times\left\\{0\right\\}}(Y\times\mathbb{P}^{1}).$

One can check that the fiber of $M_{X/Y}$ over $t=0$ is
$\mathbb{P}_{X}(C_{X/Y}\oplus\mathcal{O}_{X})\cup\operatorname{Bl}_{X}Y$ with
the two components intersecting in $\mathbb{P}(C_{X/Y})$. We then define
$M^{0}_{X/Y}$ to be the complement of $\operatorname{Bl}_{X}Y$ in $M_{X/Y}$.

######  Remark 3.3.1.

From the previous construction, we see that if $Y$ has pure dimension $d$,
then $C_{X/Y}$ has pure dimension $d$.

###  3.4. Segre classes

To each cone we can associate a cycle called its Segre class. Methods to
compute Segre classes are reviewed in Section 7.3.

######  Definition 3.4.1.

Let $X\hookrightarrow Y$ be an embedding and $C_{X/Y}$ the normal cone of $X$
in $Y$. Let $\mathcal{O}(1)$ be the tautological line bundle on

$\mathbb{P}(C_{X/Y}\oplus\mathcal{O}_{X})\stackrel{{\scriptstyle
q}}{{\longrightarrow}}X.$

We define the _Segre class_ $s(X,Y)$ of $C_{X/Y}$ to be the cycle

$s(X,Y)=q_{*}\left(\sum_{i\geq
0}c_{1}(\mathcal{O}(1))^{i}\cap[\mathbb{P}(C_{X/Y}\oplus\mathcal{O}_{X})]\right)$

######  Example 3.4.2.

If $C_{X/Y}=N_{X/Y}$ is a vector bundle, then

$s(C_{X/Y})=c(C_{X/Y})^{-1}.$

Here $c$ denotes the total Chern class. (See [Ful98, Proposition 4.1].)

######  Example 3.4.3.

Let us look again at the example in Section 2.2 and see that there is one
conic tangent to five general lines in a plane. We use [Ful98, Prop. 9.1.1]
and straightforward computations of Segre classes of vector bundles. Let
$\nu_{2}\colon\mathbb{P}^{2}\to\mathbb{P}^{5}$ be the Veronese embedding of
degree $2$. We can check that $\mathbb{P}^{2}$ parametrises the space of
double lines inside the space of conics. Both $Z$ and $\mathbb{P}^{5}$ are
smooth and the Chern class of their tangent bundle is easily computed from the
Euler sequence. Let $L$ be the class of a line in $Z$, and $N_{i}$ the
restriction of the normal bundle
$N_{Q_{i}/\mathbb{P}^{5}}\simeq\mathcal{O}_{Q_{i}}(2)$ to $Z$. We have
$c(N_{i})=(1+4L)$. The contribution of $Z$ to $\deg(\bigcap_{i=1}^{5}Q_{i})$
is then computed by

$\displaystyle(Q_{1}\cdot\ldots\cdot Q_{5}\cdot\mathbb{P}^{5})^{Z}=$
$\displaystyle\\{\prod
c(N_{i})c(T_{\mathbb{P}^{5}|Z})^{-1}c(T_{Z})\cap[Z]\\}_{0}$ $\displaystyle=$
$\displaystyle\deg((1+4L)^{5}(1+2L)^{-6}(1+L)^{3}[\mathbb{P}^{2}])$
$\displaystyle=$ $\displaystyle 31.$

From this, we see that the plane of double lines contributes $31$ to the
intersection $\cap_{i=1}^{5}Q_{i}$ and the residual scheme is just one reduced
point which represents the nondegenerate conic tangent to the five given
general lines.

######  Proposition 3.4.4.

[Ful98, Example 4.1.6,10.1.10]. Let $\mathcal{C}$ be a cone over
$X\times\mathbb{A}^{1}$ flat over $\mathbb{A}^{1}$ and let $C_{t}$ be the
restriction of $\mathcal{C}$ to $X\times\left\\{t\right\\}.$ Then

$s(C_{0})=s(C_{1})\;\in\;A_{*}(X)$

Computing Segre classes by blowing up. We recall some useful formulae from
[Ful98].

######  Proposition 3.4.5.

[Ful98, Corollary 4.2.2] Let $X$ be a proper closed sub-scheme of $Y,$
$\tilde{Y}=\operatorname{Bl}_{X}Y$ and $\eta\colon\tilde{X}\to X$ the
exceptional divisor. Then

$s(X,Y)=\sum_{k\geq 1}{(-1)^{k-1}\eta_{*}[\tilde{X}]^{k}}$

######  Lemma 3.4.6.

[Ful98, Example 8.3.9] Let $X\subseteq Y$ be non-singular varieties, and
consider $\tilde{Y}=\operatorname{Bl}_{X}Y$ with exceptional divisor
$\tilde{X}$:

${\tilde{X}}$${\tilde{Y}}$${X}$${Y}$$\scriptstyle{j}$$\scriptstyle{\eta}$$\scriptstyle{f}$$\scriptstyle{i}$

Then the ring structure on $A_{*}(\tilde{Y})$ is given by:

(i):

$f^{*}y\cdot f^{*}y^{\prime}=f^{*}(y\cdot y^{\prime})$

(ii):

$j_{*}(\tilde{x})\cdot
j_{*}(\tilde{x}^{\prime})=j_{*}\left(c_{1}(j^{*}\mathcal{O}_{\tilde{Y}}(\tilde{X}))\cdot\tilde{x}\cdot\tilde{x}^{\prime}\right)$

(iii):

$f^{*}y\cdot j_{*}(\tilde{x})=j_{*}((\eta^{*}i^{*}y)\cdot\tilde{x})$

###  3.5. Failure of exact sequences

Given a composition of regular embeddings

$X\hookrightarrow Y\hookrightarrow Z$

there exists a short exact sequence of vector bundles

$0\to N_{X/Y}\to N_{X/Z}\to N_{Y/Z}\rvert_{X}\to 0.$

When the embeddings are not regular, we still have induced maps of cones

(1) $C_{X/Y}\to C_{X/Z}\to C_{Y/Z}\rvert_{X}.$

Intuitively, a sequence of cones should be called exact if locally we have
$C_{X/Z}\simeq C_{X/Y}\times_{X}C_{Y/Z}\rvert_{X}$. See [Ful98, Example 4.1.6]
for a more restrictive definition of exactness. Sequence (1) is not exact in
general.

######  Example 3.5.1.

Exact sequences of cones

(a) Consider
$X\overset{j}{\hookrightarrow}Y\overset{i}{\hookrightarrow}Z=Y\times\mathbb{A}^{n}$,
then $C_{X/Z}\simeq C_{X/Y}\times\mathbb{A}^{n}$.

(b) Let
$X\overset{j}{\hookrightarrow}Y=X\times\mathbb{A}^{n}\overset{i}{\hookrightarrow}Z$.
Then, the sequence of cones

$0\to\mathbb{A}^{n}_{X}\to C_{X/Z}\to j^{*}C_{Y/Z}\to 0$

is not necessarily exact.

###### Proof.

(a) Locally we may assume

$\displaystyle Y=\operatorname{Spec}R$ $\displaystyle
X=\operatorname{Spec}R/I$ $\displaystyle
Z=\operatorname{Spec}R[t_{1},\ldots,t_{n}].$

Then the ideal of $X$ in $Z$ is $J=I+(t_{1},\ldots,t_{n})$, and intuitively
there are no mixed relations between the generators of $J$ and the $t_{i}$’s.
Indeed, there is a surjective map

$\bigoplus_{i=0}^{k}I^{i}/I^{i+1}\otimes_{R/I}(t_{1},\ldots,t_{n})^{k-i}\to
J^{k}/J^{k+1}$

To show injectivity, pick a homogeneous polynomial of degree $k$ in the
generators of $I$ and $t_{i}$. Suppose its image lies in $J^{k+1}$and the
degree in $t_{i}$ is $k-j$. Then, the corresponding coefficient must then lie
in $I^{j+1}$ which shows that the polynomial is zero on the left-hand side
too.

(b) A counterexample is given by $Z$ the cone over a quadric, $Y$ a line
through the vertex, and $X$ the vertex. One sees that the sequence

$\mathbb{C}[v_{0},v_{2}]\hookrightarrow\mathbb{C}[v_{0},v_{1},v_{2}]/(v_{2}^{2}-v_{0}v_{1})\twoheadrightarrow\mathbb{C}[v_{1}]$

does not split because

$\mathbb{C}[v_{0},v_{2}]\otimes_{\mathbb{C}}\mathbb{C}[v_{1}]\ncong\mathbb{C}[v_{0},v_{1},v_{2}]/(v_{2}^{2}-v_{0}v_{1}).$

See Example 3.5.3 (b) below for a proof with Segre classes. ∎

######  Remark 3.5.2.

Case (a) admits the following generalisation (see [Ful98, Theorem 6.5]). For

$X\overset{j}{\hookrightarrow}Y\overset{i}{\hookrightarrow}E,$

with $E$ a vector bundle on $Y$ and $i$ the zero section, we have

$C_{X/Z}\simeq C_{X/Y}\times_{X}j^{*}E.$

This does not generalise to $E$ any cone on $Y$.

######  Example 3.5.3.

This is expanding on [Ful98, Example 4.2.8].

(a) Consider $X\overset{j}{\hookrightarrow}Y\overset{i}{\hookrightarrow}Z$
with $i$ a regular embedding; then not necessarily $s(X,Y)=s(X,Z)\cap
c(N_{Y/Z})$.

(b) Consider $X\overset{j}{\hookrightarrow}Y\overset{i}{\hookrightarrow}Z$
with $j$ a regular embedding of codimension 1, i.e. $X\subseteq Y$ a Cartier
divisor; it is not always true that

$j^{*}s(Y,Z)=s(X,Z)\cap c(N_{X/Y}).$

###### Proof.

(a) Take $Z=\mathbb{P}^{2}$, $Y$ a nodal cubic, e.g. the one given by

$\\{f=y^{2}z-x^{3}-x^{2}z=0\\}$

and $X$ the node, which is the origin of the affine chart
$\mathbb{A}^{2}_{x,y}$. Then $Y\subseteq Z$ is a Cartier divisor with normal
bundle $\mathcal{O}_{Y}(3)$. $C_{X/Z}$ is a vector bundle over $X$ and
$s(X,Z)=[\underline{0}].$ On the other hand, $s(X,Y)=2[\underline{0}]$. This
can be seen directly from the local isomorphism

$C_{X/Y}\simeq\operatorname{Spec}\mathbb{C}[t_{1},t_{2}]/(t_{1}^{2}-t_{2}^{2}).$

The cone $\operatorname{Spec}\mathbb{C}[t_{1},t_{2}]/(t_{1}^{2}-t_{2}^{2})$
has two irreducible components (the tangent directions to the two branches of
the curve at the node).

(b) Let $Z$ be the affine cone given by

$\operatorname{Spec}\mathbb{C}[x,y,z]/(x^{2}+y^{2}-z^{2}).$

Let $X$ be the vertex of the cone and $Y=\\{x=y-z=0\\}$ a line through $X$.
Then the normal cone $C_{Y/Z}$ is given by

$\mathbb{C}[x,y,z][w_{0},w_{1}]/(x,y-z,xw_{0}+(y+z)w_{1}).$

Let us compute the Segre class separately of $C_{Y/Z}$. When restricting to
the point, only the degree 1 part of $s(Y,Z)$ matters, hence we may discard
the component supported on $\\{x=y=0\\}$. Since the remaining component is a
vector bundle on $Y$, we get $j^{*}s(Y,Z)=[X]$.

Let us compute $s(X,Z)$ by means of formula 3.4.5. Consider
$\tilde{Z}=\operatorname{Bl}_{X}Z$ and $\tilde{X}$ the exceptional divisor.
$Z$ has an $A_{1}$-singularity at $X$ and $\tilde{Z}$ is the minimal
resolution. This shows that $\tilde{X}$ is a $(-2)$-curve and thus

$s(X,Z)=\pi_{*}([\tilde{X}]-\tilde{X}\cdot[\tilde{X}])=0-(-2[X])$

We thus get $j^{*}s(Y,Z)\neq s(X,Z)\cap c(N_{X/Y})$. ∎

##  4\. Virtual pull-backs

In order to define the virtual fundamental class, we need two more
ingredients: the Gysin map and a vector bundle containing our cone. We define
virtual classes in a simple context and we give examples. The main references
are [Ful98], Chapters 5 and 6.

######  Proposition 4.0.1 ([Ful98, Theorem 3.3]).

Let $E$ be a vector bundle of rank $r$ on $X$, then the flat pull-back
$\pi^{*}\colon A_{k-r}(X)\to A_{k}(E)$ is an isomorphism for any $k$.

######  Notation 4.0.2.

Let $0_{E}^{!}\colon A_{k}(E)\to A_{k-r}(X)$ denote the inverse of the flat
pull-back $\pi^{*}\colon A_{k-r}(X)\to A_{k}(E)$.

######  Construction 4.0.3.

Suppose that we have the following data:

  1. (1)

an embedding $X\hookrightarrow Y$, and

  2. (2)

a vector bundle $E_{X/Y}$ of rank $r$ containing the cone
$C_{X/Y}\hookrightarrow E_{X/Y}$.

Then, we have morphisms:

(2) $\displaystyle
A_{*}(Y)\xrightarrow{\sigma}A_{*}(C_{X/Y})\xrightarrow{i_{*}}A_{*}(E_{X/Y})\xrightarrow{0_{E}^{!}}A_{*-r}(X)$

where the first arrow is defined as $\sigma[V]=[C_{V\cap X}V]$, the second one
is the proper push-forward, and the third one is the morphism in (4.0.2). We
denote the composition in (2) by $f^{!}_{E_{X/Y}}$ and we call it a
generalised Gysin map (or a virtual pull-back). Suppose furthermore that $Y$
is pure-dimensional. W define the virtual fundamental class of $X$ with
respect to the obstruction theory $E_{X/Y}$ to be the class

$[X]_{E_{X/Y}}^{\mathrm{vir}}=f^{!}_{E_{X/Y}}[Y]=0^{!}_{E}[C_{X/Y}].$

######  Definition 4.0.4.

We call $E_{X/Y}$ with property (2) an _obstruction bundle_ for
$X\hookrightarrow Y$.

The following formula is useful for computing virtual classes.

######  Proposition 4.0.5.

[Ful98, Example 4.1.8] If $C_{X/Y}$ is a cone of pure dimension $k$ and
$E_{X/Y}$ a rank $r$ vector bundle containing it, we have that

$0^{!}_{E}[C_{X/Y}]=\left\\{c(E_{X/Y})\cap
s(X,Y)\right\\}_{k-r}=\sum_{i=0}^{r}{c_{i}(E)\cap s_{k-r+i}(C)}$

where $c(E_{X/Y})$ denotes the total Chern class of the vector bundle
$E_{X/Y}$.

######  Example 4.0.6.

Let $X\hookrightarrow Y$ be a regular embedding with $Y$ pure-dimensional,
then $C_{X/Y}=N_{X/Y}$ is a vector bundle. If $E_{X/Y}$ is an obstruction
bundle, we may look at the following exact sequence of vector bundles

$0\to N_{X/Y}\to E_{X/Y}\to E\to 0.$

$E$ is called an _excess bundle_. As the Segre class of a vector bundle is the
inverse of its total Chern class (Example 3.4.2), from the above exact
sequence we get that the virtual class is given by

$[X]_{E_{X/Y}}^{\mathrm{vir}}=c_{top}(E)\cap[X].$

###  4.1. Obstruction bundles from global generators

In order to define a virtual class on $X$, we need to embed $X$ in a pure-
dimensional $Y$ and find a vector bundle $E_{X/Y}$ which contains the normal
cone $C_{X/Y}$. Let us now construct such bundles. Given an embedding
$i:X\hookrightarrow Y$ of schemes and $f_{1},\ldots,f_{n}$ generators of the
ideal of $X$ in $Y$, we have a surjective morphism

$\bigoplus_{j=1}^{r}O_{Y}(-D_{j})\xrightarrow{(f_{1},\dots,f_{r})}\mathcal{I}_{X/Y}\to
0$

where $D_{j}$ are Cartier divisors corresponding to $f_{j}$. Pulling this
sequence back to $X$, we get an induced embedding of cones

$C_{X/Y}\hookrightarrow\operatorname{Spec}(\mathcal{S}ym(\mathcal{I}_{X/Y}^{\vee}/\mathcal{I}_{X/Y}^{\vee
2}))\hookrightarrow\operatorname{Spec}(\mathcal{S}ym(\bigoplus_{1}^{r}\mathcal{O}_{X}(i^{*}D_{j}))).$

We define

$E_{X/Y}:=\operatorname{Spec}(\mathcal{S}ym(\bigoplus_{1}^{r}\mathcal{O}_{X}(i^{*}D_{j}))),$

and we call it an obstruction bundle for $i$.

######  Example 4.1.1.

Let us compute the virtual class of $X=V(x^{2},xy)\subset\mathbb{P}^{2}$,
induced by the vector bundle
$E=\mathcal{O}_{\mathbb{P}^{2}}(2)\oplus\mathcal{O}_{\mathbb{P}^{2}}(2)$.

To see that this makes sense, note that $X=\\{s=0\\}$ for $s$ a section of
$E$, which shows that $E^{\vee}\rvert_{X}\to I/I^{2}\to 0$. Let $0_{E}:X\to E$
denote the zero section of $E$. We define the virtual class of $X$ by

$[X]^{\mathrm{vir}}=0_{E}^{!}([C_{X/\mathbb{P}^{2}}]).$

By Example 3.1.2, $[C_{X/\mathbb{P}^{2}}]=[C_{0}]+[L]$, where $C_{0}$ denotes
the component supported on the origin and $L$ denotes the component of
$C_{X/\mathbb{P}^{2}}$ supported on the $y$-axis. Since
$0_{E}^{!}([C_{\mathbb{P}^{2}}X])=0_{E}^{!}([C_{0}])+0_{E}^{!}([L])$ we may
deal with components of the cone separately.

Note that $C_{0}$ is the scheme given by the ring

$\mathbb{C}[x,y,A,B]/(x^{2},xy,y^{2},yA-xB).$

Its reduced structure is the plane given by the ring $\mathbb{C}[A,B]$, and
its multiplicity is two. This gives

(3) $0_{E}^{!}([C_{0}])=0_{E}^{!}(\pi^{*}2[\underline{0}])=2[\underline{0}].$

We now compute $0_{E}^{!}([L])$. In the chart $z\neq 0$, where the singular
point is,

$L\subseteq C_{X/\mathbb{P}^{2}}$

is given by $x=B=0$. This shows that $L$ is a line bundle locally defined by
$B=0$. The normal bundle of the line $x=0$ is the line bundle
$\mathcal{O}_{\mathbb{P}^{1}}(1)$ and the line bundle $L$ is isomorphic to
$\mathcal{O}_{\mathbb{P}^{1}}(1)$ away from the origin. The embedding of $L$
in $\mathcal{O}_{\mathbb{P}^{1}}(2)\oplus\mathcal{O}_{\mathbb{P}^{1}}(2)$ is
given by the injective morphism of sheaves

$0\to\mathcal{O}_{\mathbb{P}^{1}}(1)\stackrel{{\scriptstyle(x,y)}}{{\longrightarrow}}\mathcal{O}_{\mathbb{P}^{1}}(2)\oplus\mathcal{O}_{\mathbb{P}^{1}}(2).$

More precisely, $L$ is the closure of the image of
$\mathcal{O}_{\mathbb{P}^{1}}(1)$ inside
$\mathcal{O}_{\mathbb{P}^{1}}(2)\oplus\mathcal{O}_{\mathbb{P}^{1}}(2)$. Since
on the line $x=0$, the above morphism is multiplication with $(0,y)$, we have
that $L=\mathcal{O}_{\mathbb{P}^{1}}(2)$. By Proposition 4.0.5, we obtain

(4) $0_{E}^{!}([L])=\left\\{c(E)\cap c(L)^{-1}\right\\}_{0}=c_{1}(E/L)=2[P],$

where $[P]$ is the class of any point on $X^{\text{red}}\simeq\mathbb{P}^{1}$.
By (3) and (4), we have that

$[X]^{\mathrm{vir}}=4[P].$

######  Example 4.1.2.

This is [Ful98, Example 4.2.2]. Consider a smooth surface $Y$, and let $A,B$
and $D$ be effective Cartier divisors on $Y$, such that $A$ and $B$ meet
transversally at a smooth point $P$ away from $D$. Consider $A^{\prime}=A+D$
and $B^{\prime}=B+D$ and let $X$ be the scheme theoretic intersection of
$A^{\prime}$ and $B^{\prime}$. Let us compute the Segre class and the virtual
class of $X$ induced by
$E_{X/Y}=\mathcal{O}_{Y}(A^{\prime})\oplus\mathcal{O}_{Y}(B^{\prime})$. Note
that $A^{\prime}$ and $B^{\prime}$ correspond to generators for
$\mathcal{I}_{X/Y},$ and $E_{X/Y}$ is an obstruction bundle for $X\to Y$.

We compute the Segre class after blowing up, as in Proposition 3.4.5. Let
$\tilde{Y}$ be the blow-up of $Y$ at $P$, let $\tilde{D}$ be the strict
transform of $D$ and let $E$ be the exceptional divisor. $\tilde{Y}$ is
isomorphic to the blow up of $Y$ along $X$ and under this isomorphism
$\tilde{X}$ corresponds to $\tilde{D}+E$. Recall that
$s(X,Y)=\pi_{*}([\tilde{X}]-\tilde{X}\cdot[\tilde{X}])$, hence we need to
compute $(\tilde{D}+E)^{2}$. Using the formulas in Lemma 3.4.6, we obtain

$\tilde{D}^{2}=D^{2}\quad\text{and}\quad\tilde{D}\cdot E=0\quad\text{and}\quad
E^{2}=-[p_{E}].$

This is because the blow-up does not change a neighbourhood of $D$ and
$\tilde{D}$ and $E$ do not intersect. Then,

$s(X,Y)=\pi_{*}([\tilde{D}+E]-(\tilde{D}+E)\cdot[\tilde{D}+E])=[D]-D\cdot[D]+[P].$

The virtual class on $X$ is given by

$\displaystyle[X]^{\text{vir}}$ $\displaystyle=\\{c(E_{X/Y})\cap
s(X,Y)\\}_{0}$ $\displaystyle=c_{1}(E_{X/Y})\cap[D]-D\cdot[D]+[P]$
$\displaystyle=(D+A+B)\cdot[D]+[P].$

######  Example 4.1.3.

Let us now look at a smooth surface $Y$, and $A,B,D$ effective Cartier
divisors on $Y$ as before, but now let $A$ and $B$ meet transversally at a
smooth point $P\in D$. For example, consider $Y=\mathbb{P}^{2}$ with
homogeneous coordinates $[x:y:z]$, $A=\\{y=0\\}$, $B=\\{x-y=0\\}$,
$D=\\{x=0\\}$. We then have that the ideal of $A^{\prime}\cap B^{\prime}$ is
$(xy,x(x-y))$ or equivalently $(xy,x^{2})$. This shows that $A^{\prime}\cap
B^{\prime}$ is isomorphic to $X$ in Example 4.1.1. Although the set theoretic
intersection of $A^{\prime}$ and $B^{\prime}$ is $D$, the scheme theoretic
intersection is different and the virtual class is different from $D$. This is
compatible with the fact that we can deform $A$ and $B$ in Example 4.1.2 such
that in the limit $P\in D$ and virtual classes are deformation invariant. We
can easily check deformation invariance in this case. If we take $A,B,D$ in
example 4.1.2 to be lines, we get $[X]^{\mathrm{vir}}=4[P]$. This agrees with
Example 4.1.1.

##  5\. Gluing obstruction bundles

In this section we motivate and define obstruction theories. We give the
general definition of virtual classes as it appears in [BF97]. For many
aspects in [BF97] which we do not discuss here, we refer to the original paper
by Behrend and Fantechi. We refer to [Kre99] and [Man12a] for details
regarding the stacky version of virtual pull-backs.

For what we have seen so far, it is often easy to define a virtual class on a
scheme. What is difficult is to define a virtual class of a given _expected
dimension_ (see 8.1.1). Recall that the dimension of the virtual cycle
$[X]_{E_{X/Y}}^{\mathrm{vir}}$ depends on the rank of the obstruction bundle
$E_{X/Y}$. Obstruction bundles induced by generators of an ideal as in Section
4.1 may often lead to trivial virtual classes. It is then natural to look for
obstruction bundles of lower rank. Let us see what can go wrong in an example.

######  Example 5.0.1.

Consider the embedding of the twisted cubic

${\mathbb{P}^{1}}$${\mathbb{P}^{3}}$${\left[s:t\right]}$${\left[s^{3}:s^{2}t:st^{2}:t^{3}\right]}$

with ideal $I_{C/\mathbb{P}^{3}}$ generated by $(xz-y^{2},yw-z^{2},xw-yz)$.
Let $A=xz-y^{2}$, $B=yw-z^{2}$, $C=xw-yz$. According to the previous
construction, the obstruction bundle restricted to $C$ is
$E_{C/\mathbb{P}^{3}}=\oplus_{1}^{3}\mathcal{O}_{\mathbb{P}^{1}}(6)$. On the
other hand, $C$ is smooth, so we can take the normal bundle as obstruction
bundle. In the following, we construct the normal bundle from equations and
gluing data. We see that we have two relations among the generators of
$I_{C/\mathbb{P}^{3}}$. Let us pick one of them, for example $wA+By-zC=0$.
This relation induces a morphism

$\oplus_{1}^{3}\mathcal{O}_{\mathbb{P}^{1}}(6)\to\mathcal{O}_{\mathbb{P}^{1}}(9),$

given by diagonal multiplication with $(w,y,-z)$. The kernel of this morphism
is $N_{C/\mathbb{P}^{3}}$. The cokernel is $\mathbb{C}_{P}$, where
$P=[1:0:0:0]$. This shows that we have an exact sequence

$0\to
N_{C/\mathbb{P}^{3}}\to\oplus_{1}^{3}\mathcal{O}_{\mathbb{P}^{1}}(6)\to\mathcal{O}_{\mathbb{P}^{1}}(8)\to
0.$

One can check that
$N_{C/\mathbb{P}^{3}}\simeq\mathcal{O}_{\mathbb{P}^{1}}(5)\oplus\mathcal{O}_{\mathbb{P}^{1}}(5)$,
but we will not need this here. By the excess intersection formula in Example
4.0.6, the virtual class with respect to $E_{C/\mathbb{P}^{3}}$ is

$[C]^{\mathrm{vir}}_{E_{C/\mathbb{P}^{3}}}=8[pt].$

Taking the obstruction bundle of $C$ in $\mathbb{P}^{3}$ equal to
$N_{C/\mathbb{P}^{3}}$, we get $[C]^{\mathrm{vir}}=[C]$.

######  Example 5.0.2.

Let us look at one scheme which is not smooth. In this case an obstruction
bundle is not obvious. Consider $\mathbb{P}^{5}\times\mathbb{P}^{4}$ with
coordinates $\left([y_{0},\ldots,y_{4},\xi],[x_{0},\ldots,x_{4}]\right)$ and
$X\subset\mathbb{P}^{5}\times\mathbb{P}^{4}$, with equations

$\displaystyle y_{0}\xi,\ldots,y_{4}\xi$ $\displaystyle
x^{s}_{0}y_{0}+\dots+x^{s}_{4}y_{4}$ $\displaystyle Q_{1},\ldots,Q_{l}$

with $Q_{j}$ homogeneous polynomials in $x_{0},\ldots,x_{4}$. Since we do not
impose any conditions on $Q_{i}$, $X$ can be very singular and may have many
components, so the cone $C_{X/\mathbb{P}^{5}\times\mathbb{P}^{4}}$ may not be
a bundle as we had in the previous example. As before, we have an obstruction
bundle $E$ given by global generators $A_{i}=\xi y_{i}$,
$B=x^{s}_{0}y_{0}+\dots x^{s}_{4}y_{4}$, $C_{j}=Q_{j},$ but we can find a
smaller one in the following way. Let $L$ be the line bundle
$\mathcal{O}_{\mathbb{P}^{5}\times\mathbb{P}^{4}}(2,s).$ Then we have a
morphism

$\displaystyle E$ $\displaystyle\to L$ $\displaystyle(A,B,C)$
$\displaystyle\mapsto x^{s}_{0}A_{0}+\ldots+x^{s}_{4}A_{4}-\xi B$

Since not all $x_{i}$ are zero, this morphism is surjective. Let $E^{\prime}$
denote the kernel of $E\to L$. Since the equation
$x^{s}_{0}A_{0}+\ldots+x^{s}_{4}A_{4}-\xi B$ vanishes on
$C_{X/\mathbb{P}^{5}\times\mathbb{P}^{4}}$, we have that $E^{\prime}$ is an
obstruction bundle for $X\to\mathbb{P}^{5}\times\mathbb{P}^{4}$. $E^{\prime}$
is sometimes called a reduced obstruction theory.

Let us now give a new definition of obstruction bundles motivated by the
previous examples. The following definition is less coarse than the one in
4.1, but still rather naïve.

######  Definition 5.0.3.

Let $X$ be a DM stack (or a scheme). An obstruction theory $E_{X/V}$ on $X$ is
given by the following data:

  1. (1)

a global embedding $X\hookrightarrow V$, with $V$ a smooth stack,

  2. (2)

a covering of $V$ by open sets $V_{i}$, and on each $U_{i}=X\cap V_{i}$ a set
of generators for the restriction to $U_{i}$ of the ideal $I_{U_{i}/V_{i}}$;
these induce obstruction bundles $E_{i}$,

  3. (3)

gluing data for the sets of generators, i.e. isomorphisms

$\phi_{ij}:E_{i}|_{U_{ij}}\to E_{j}|_{U_{ij}}$

such that $\phi_{ij}$ satisfy cocycle and such that the following diagram
commutes

$\textstyle{C_{U_{i}/V_{i}}|_{U_{ij}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\phi}_{ij}}$$\textstyle{E_{i}|_{U_{ij}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{ij}}$$\textstyle{C_{U_{j}/V_{j}}|_{U_{ij}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{j}|_{U_{ij}},}$

Here $\bar{\phi}_{ij}$ is the canonical isomorphism, and the horizontal maps
are the embeddings of the cones in the corresponding obstruction theories.

######  Remark 5.0.4.

Given $X$ in $V$ as above and equations of $X$ in $V$, everything in the
definition above is explicit. We have seen that cones can be described
explicitly inside the corresponding obstruction bundles and gluing data can be
given in terms of relations between generators of $I_{U_{i}/V_{i}}$ and
$I_{U_{j}/V_{j}}$.

In conclusion, if we want to get reasonable virtual cycles, we need a lot of
information: local equations of our space in a smooth ambient space, and
gluing data for local obstruction bundles. In this way computations get
quickly very messy. The information we want can be encoded more efficiently in
a stacky gadget that we describe below. For this, we remark that given the
embedding $X\hookrightarrow Y$, the naïve obstruction bundle
$E_{X/Y}=\bigoplus\mathcal{O}_{Y}(D_{i})$ comes with a piece of extra data we
did not use so far. We have a morphism

(5) $T_{Y}\rvert_{X}\xrightarrow{(\delta f_{i})}E_{X/Y},$

defining an action of the tangent space of the ambient restricted to $X$. This
motivates the following.

Let $X$ be a scheme (DM stack). In the following $\mathbb{L}^{\bullet}_{X}\in
D^{\leq 0}(X)$ denotes the cotangent complex of $X$, and
$\mathbb{L}^{\geq-1}_{X/Y}$ the $[-1,0]$ truncation of
$\mathbb{L}^{\bullet}_{X}$. See Remark 5.0.12 for examples of truncated
cotangent complexes.

######  Definition 5.0.5.

[BF97, Definition 4.4] A perfect obstruction theory for $X$ is a complex
$E^{\bullet}_{X}\in D^{[-1,0]}(X)$ which is locally quasi-isomorphic to a
complex of locally free sheaves, and admits a morphism
$E^{\bullet}_{X}\xrightarrow{\phi}\mathbb{L}^{\geq-1}_{X}$ such that

  1. (1)

$h^{0}(\phi)\colon h^{0}(E^{\bullet}_{X})\to
h^{0}(\mathbb{L}^{\geq-1}_{X})\;\;\;\;\text{ is an isomorphism}$

  2. (2)

$h^{-1}(\phi)\colon h^{-1}(E^{\bullet}_{X})\to
h^{-1}(\mathbb{L}^{\geq-1}_{X})\;\;\;\;\text{ is surjective}$.

In particular, we get a perfect obstruction theory if locally on $X$ we may
find embeddings $U\subseteq V$, with $V$ smooth, together with obstruction
bundles $E_{U/V}$, such that the local tangent-obstruction complexes
$[T_{V}|_{U}\to E_{U/V}]$ glue to $E^{\bullet}_{X}$, which is a global object
on $X$. Such a complex gives rise to an algebraic stack denoted
$h^{1}/h^{0}(E^{\bullet}_{X})$ such that

$h^{1}/h^{0}(E^{\bullet}_{X})|_{U}\simeq E_{U/V}/T_{V}|_{U}.$

See [BF97, §2] for the definition. We use the notation
$\mathfrak{E}_{X}=h^{1}/h^{0}(E^{\bullet}_{X})$. We call $\mathfrak{E}_{X}$ a
_vector bundle stack_. The rank of a vector bundle stack
$h^{1}/h^{0}(E^{\bullet}_{X})$ as above, is defined to be
$\operatorname{rk}(h^{1}(E^{\bullet}_{X}))-\operatorname{rk}(h^{0}(E^{\bullet}_{X}))$.
Notice that this number may be negative.

We have an analogous construction for cones. Given local charts $U_{i}$ of our
space $X$ which embed into possibly unrelated smooth ambient spaces $V_{i}$,
the action in (5) preserves the cone $C_{U/V}$ and we have a stack
$\mathfrak{C}_{X}$ such that

$\mathfrak{C}_{X}|_{U}\simeq C_{U/V}/T_{V}|_{U}.$

The stack $\mathfrak{C}_{X}$ is called _the intrinsic normal cone_ of $X$. It
is a zero dimensional stack, independent of embeddings in smooth stacks. See
[BF97] for the construction.

Conditions (1) and (2) in Definition 5.0.5 are equivalent to the fact that the
vector bundle stack $\mathfrak{E}_{X}$ associated to $E^{\bullet}_{X}$
contains the intrinsic normal cone $\mathfrak{C}_{X}$ [BF97, Theorem 4.5].

With these in place, we can now define a more general virtual class.

######  Proposition 5.0.6.

[Kre99] If $\mathfrak{E}$ is a vector bundle stack on a scheme (or DM stack)
$X$, the inverse of the flat pull-back is an isomophism

$0^{!}_{\mathfrak{E}}\colon
A_{*}(\mathfrak{E})\xrightarrow{\cong}A_{*-rk(\mathfrak{E})}(X).$

######  Construction 5.0.7.

[BF97] Given a perfect obstruction theory $E^{\bullet}_{X}$ on a DM stack $X$,
we may define its virtual class by means of

$[X]^{\mathrm{vir}}_{\mathfrak{E}_{X}}=0^{!}_{\mathfrak{E}_{X}}[\mathfrak{C}_{X}].$

######  Remark 5.0.8.

Unlike Construction 4.0.3, the construction in [BF97] only requires local
embeddings in a smooth stack. This is a major advantage over Construction
4.0.3, as global embeddings in smooth stacks are sometimes unavailable.
However, Construction 5.0.7 is in general impractical for computational
purposes. For this reason, in this note we will mainly work with the more
naïve Construction 4.0.3.

######  Remark 5.0.9.

Construction 5.0.7 may be generalised to any morphism of DM type $f\colon X\to
Y$ between algebraic stacks. The relative intrinsic normal cone may locally be
constructed as follows: choose a chart $U$ for $X$, embedded in a space $V$
smooth over $Y$ (e.g. embed $U$ in a smooth $M$ and take $V=Y\times M$), then
$\mathfrak{C}_{X/Y}|_{U}=[C_{U/V}/T_{V/Y}|_{U}]$. This leads to the definition
of virtual pull-back [Man12a]:

$f^{!}_{\mathfrak{E}_{X/Y}}\colon A_{*}(Y)\to
A_{*-\operatorname{rk}(\mathfrak{E}_{X/Y})}(X).$

It naturally endows $X$ with a virtual class whenever $Y$ is pure-dimensional.

######  Remark 5.0.10.

We are usually interested in finding obstruction theories for moduli functors.
In many cases obstruction theories are available from deformation theory. More
precisely, consider the DM stack $X^{F}$ representing a certain moduli functor
$F$. In this case, the study of the deformation theory of $F$ always gives a
local tangent-obstruction theory, i.e. the Kuranishi map:

$T^{1}_{F}\to T^{2}_{F}.$

A perfect obstruction theory is usually given considering a global version of
this picture, see the examples in [Beh97]. Similar constructions can be done
for other moduli spaces (quasimap spaces [CFKM14], certain moduli spaces of
sheaves [Tho00] or stable pairs on Calabi-Yau 3-folds [PT09]). See [PT14] for
an overview.

######  Remark 5.0.11.

Above we have constructed $\mathfrak{E}_{X}$ from a two-term complex
$E^{\bullet}_{X}\in D^{[-1,0]}(X)$. There is a more general construction due
to Chang and Li (see [CL11]). Their ingredients are a semi-perfect obstruction
theory and a Gysin map from the obstruction sheaf. A semi-perfect obstruction
theory consists of local obstruction theories and gluing data for the
obstruction sheaf.

######  Remark 5.0.12.

We conclude this section by describing the cotangent complex in some simple
cases.

  1. (1)

If $X\hookrightarrow Y$ is an embedding, then

$\mathbb{L}^{\geq-1}_{X/Y}=[\mathcal{I}/\mathcal{I}^{2}\to 0].$

In this case, any vector bundle $E$ which surjects onto
$\mathcal{I}/\mathcal{I}^{2}$ gives a perfect obstruction theory. Many of the
examples in this note are of this type.

  2. (2)

Let $Y$ be a point and $X$ a smooth DM stack, then
$\mathbb{L}^{\bullet}_{X/pt}\cong\Omega_{X}$ and given any vector bundle $E$,
the complex

$[E\xrightarrow{0}\Omega_{X}]$

is a perfect obstruction theory. Notice that the resulting virtual class is
$[X]^{\mathrm{vir}}=c_{\text{top}}(E)\cap[X]$.

  3. (3)

Let $g$ be an embedding of $X\hookrightarrow Y\times M$ relative to $Y$, with
$M$ smooth, then the truncated relative cotangent complex has the following
form

$\mathbb{L}^{\geq-1}_{X/Y}=[\mathcal{I}/\mathcal{I}^{2}\to\Omega_{M\times
Y/Y}]$

where $\mathcal{I}$ is the ideal of the embedding. If $X$ is l.c.i. in
$Y\times M$ then the above is a two-term complex of vector bundles, and it
represents the full cotangent complex $\mathbb{L}^{\bullet}_{X/Y}$.

##  6\. Deformations of obstructions and cones

In this section we discuss functoriality of virtual pull-backs. We give some
elementary examples. The main references for the general theory are [KKP03]
and [Man12a].

######  Construction 6.0.1.

(Deformation of vector bundles) Let

$0\rightarrow E\xrightarrow{\phi}F\rightarrow G\rightarrow 0$

be a sequence of vector bundles on a scheme/ stack X. Consider the family of
morphisms

$0\rightarrow E\xrightarrow{(t\cdot id_{E},\phi)}E\oplus F$

on $X\times\mathbb{A}^{1}$. The Cokernel $\operatorname{coker}((t\cdot
id_{E},\phi)$ is a family of vector bundles

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3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-26.09778pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{A}^{1}}$}}}}}}}\ignorespaces}}}}\ignorespaces\;\;\;\text{s.t.}\;\;\mathcal{E}\rvert_{\mathbb{A}^{1}\setminus\left\\{0\right\\}}\cong
F\times\mathbb{A}^{1}\setminus\left\\{0\right\\}\;\;\;\text{and}\;\;\;\;\pi^{-1}(0)=E\oplus
G.$

Given a cone $C\subseteq F$, considering the closure of
$C\times\mathbb{A}^{1}\setminus\left\\{0\right\\}$ in $\mathfrak{E}$, we get a
central fibre $C_{0}$ which we call the limit of $C$ in $E\oplus G$. By
construction, the limit $C_{0}$ is rationally equivalent to $C$.

######  Proposition 6.0.2.

Given embeddings $X\hookrightarrow Y\hookrightarrow Z$ and an exact sequence
of obstruction bundles

$0\to E_{X/Y}\to E_{X/Z}\to E_{Y/Z}\rvert_{X}\to 0$

compatible with the maps from the normal sheaves, the virtual classes on $X$
given by

$C_{X/Z}\hookrightarrow E_{X/Z}\qquad\text{and}\qquad
C_{X/C_{Y/Z}}\hookrightarrow E_{X/Y}\oplus E_{Y/Z}\rvert_{X}$

coincide.

###### Proof.

(Sketch) Consider first the deformation to the normal cone for the map
$Y\hookrightarrow Z$. This is a flat one-parameter family
$M^{0}_{Y/Z}\to\mathbb{P}_{t}^{1}$, such that the fibers over $t\neq 0$ are
isomorphic to $Z$, and the fiber over $0$ is isomorphic to $C_{Y/Z}$ (see
section 3.3). Construct now the double deformation space

$M_{s,t}=M^{0}_{{X\times\mathbb{P}^{1}_{t}}/M^{0}_{Y/Z}}\to\mathbb{P}^{1}_{t}\times\mathbb{P}^{1}_{s}.$

This is a flat family over $\mathbb{P}^{1}_{s}$, with general fibre over
$s\neq 0$, the deformation space $M^{0}_{Y/Z}$ and special fibre over $s=0$,
the cone $C_{X\times\mathbb{P}^{1}_{t}/M^{0}_{Y/Z}}$. One can check that the
fibres of $M_{s,t}$ over $\mathbb{P}^{1}_{t}\times\mathbb{P}^{1}_{s}$ are as
follows:

  1. (1)

the fibre over $s,t\neq 0$ is $Z$;

  2. (2)

the family over $s\neq 0$ fixed, when $t$ varies is $M_{s_{0},t}=M^{0}_{Y/Z}$;
this family is flat over $\mathbb{P}^{1}_{t}$;

  3. (3)

the family over $s=0$, when $t$ varies is the one parameter family
$M_{t}=C_{X\times\mathbb{P}^{1}_{t}/M^{0}_{Y/Z}}$; this family over
$\mathbb{P}^{1}_{t}$ is not flat;

  4. (4)

the family over $t\neq 0$, when $s$ varies is $M_{s,t_{0}}=M^{0}_{X/Z}$; this
family is flat over $\mathbb{P}^{1}_{s}$;

  5. (5)

the family over $t=0$, when $s$ varies is the one parameter family $M_{s}$;
$M_{s}$ contains the flat family $M^{0}_{X/C_{Y/Z}}\to\mathbb{P}^{1}_{s}$ and
it agrees with it generically; this family is not flat.

Let $t_{0}:0\to\mathbb{P}^{1}_{t}$ be the embedding of the point $0$ and
$s_{0}:0\to\mathbb{P}^{1}_{s}$ be the embedding of the point $0$.

Let us now look at
$M_{t}=C_{X\times\mathbb{P}^{1}_{t}/M^{0}_{Y/Z}}\to\mathbb{P}^{1}_{t}$. We
have that

$t_{0}^{!}[M_{t}]=t_{1}^{!}[M_{t}]$

in $A_{*}(M_{t})$. Since the general fibre of $M_{t}\to\mathbb{P}^{1}_{t}$ is
$C_{X/Z}$, we have that

(6) $t^{!}_{1}[M_{t}]=[C_{X/Z}].$

In the following we show that $t_{0}^{!}[M_{t}]=[C_{X/C_{Y/Z}}]$ 111The fibre
over $t=0$ contains $C_{X/C_{Y/Z}}$, but it may be larger in general, so we
need a more elaborate argument..

By commutativity of intersection products, we have that

(7) $s_{0}^{!}t^{!}_{0}[M_{s,t}]=t_{0}^{!}s^{!}_{0}[M_{s,t}]$

in $A_{*}(M_{0,0})$.

One can see from the construction of $M_{s,t}$, and point (2) in the
description of the fibres of $M_{s,t}$ above, that

(8) $t^{!}_{0}[M_{s,t}]=[M^{\prime}_{s}]\quad\text{and}\quad
s^{!}_{0}[M_{s,t}]=[M_{t}].$

Here $M^{\prime}_{s}$ is obtained by discarding components of $M_{s,t}$ which
do not dominate $\mathbb{P}^{1}_{t}$, followed by restriction to $t=0$. Since
the map $M_{s,t}\to\mathbb{P}^{1}_{t}$ is generically trivial, such components
are embedded in the fibre over $t=0$. Suppose now that we have components of
$M_{s,t}$ in the fibre $t=0$, but not over $(s,t)=(0,0)$. Then, we have that a
generic fibre $M_{s,t_{0}}$, with $t_{0}\neq 0$, contains the restriction of
such a component. This contradicts point (4) in the above description of
fibres of $M_{s,t}$. Since $s_{0}^{!}$ discards components of $M_{s,t}$ which
map to $s=0$, the above discussion shows that

(9) $s^{!}_{0}[M_{s}]=s^{!}_{0}[M^{\prime}_{s}].$

Since $M_{s}$ contains $M^{0}_{X/C_{Y/Z}}$, which is flat over
$\mathbb{P}^{1}_{s}$, we have that

(10) $s^{!}_{0}[M_{s}]=s^{!}_{0}[M^{0}_{X/C_{Y/Z}}]=[C_{X/C_{Y_{Z}}}]$

in $A_{*}(M_{0,0})$. By (7), (8), (9) and (10), we have that
$t_{0}^{!}[M_{t}]=[C_{X/C_{Y/Z}}]$ in $A_{*}(M_{0,0})$. This shows that we
have a rational equivalence

(11) $[C_{X/Z}]\simeq[C_{X/C_{Y/Z}}]$

in $A_{*}(C_{X\times\mathbb{P}^{1}_{t}/M^{0}_{Y/Z}})$.

The hypothesis implies that we have an embedding

$M_{t}\hookrightarrow\mathcal{E},$

where $\mathcal{E}$ is the deformation of obstruction bundles
$E_{X/Z}\rightsquigarrow E_{X/Y}\oplus E_{Y/Z}\rvert_{X}$ (see [KKP03]).
Pushing this equivalence forward to $\mathcal{E}$ – the deformation of
obstruction bundles $E_{X/Z}\rightsquigarrow E_{X/Y}\oplus E_{Y/Z}\rvert_{X}$
we get that (11) holds in $A_{*}(\mathcal{E})$. The claim on virtual classes
follows from commutativity of $0^{!}_{\mathcal{E}}$ and restriction to
$t\in\mathbb{P}^{1}$.

∎

######  Example 6.0.3.

Let us compute the virtual class of $X=V(xz,yz)\subseteq\mathbb{P}^{3}$. The
embedding of $X$ in $\mathbb{P}^{3}$ factors through $Y=V(xz)$. This gives an
exact sequence of compatible obstruction theories:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{O}_{X}(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
85.13449pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern
1.0pt\raise 0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 85.13449pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{O}_{X}(2)\oplus\mathcal{O}_{X}(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
177.53593pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern
1.0pt\raise 0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{O}_{X}(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
234.30289pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern
1.0pt\raise 0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 234.30289pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern-3.0pt\raise-40.66444pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
32.22865pt\raise-40.66444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{E_{X/Y}}$}}}}}}}{\hbox{\kern
105.37816pt\raise-40.66444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{E_{X/\mathbb{P}^{3}}}$}}}}}}}{\hbox{\kern
176.40347pt\raise-40.66444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{E_{Y/\mathbb{P}^{3}}\rvert_{X}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

By Proposition 6.0.2, the virtual class is given by the embedding

$C_{X/C_{Y/\mathbb{P}^{3}}}\rvert_{X}\hookrightarrow E_{X/Y}\oplus
E_{Y/\mathbb{P}^{3}}.$

More precisely, we have

$[X]^{\mathrm{vir}}=0^{!}_{E_{X/Y}\oplus
E_{Y/\mathbb{P}^{3}}}[C_{X/C_{Y/\mathbb{P}^{3}}}].$

Note that $C_{Y/\mathbb{P}^{3}}\simeq\mathcal{O}_{Y}(2)$, so

$C_{X/C_{Y/\mathbb{P}^{3}}}\cong
C_{X/Y}\times_{X}C_{Y/\mathbb{P}^{3}}\rvert_{X}\cong
C_{X/Y}\oplus\mathcal{O}_{X}(2).$

Moreover, as $Y=Y_{1}\cup Y_{2}=V(z)\cup V(x)$

$C_{X/Y}\cong C_{X_{1}/Y_{1}}\cup C_{X_{2}/Y_{2}}=C_{H/H}\oplus C_{L_{1}\cup
L_{2}/Y_{1}},$

where

$X_{1}=X\times_{Y}Y_{1}=H=V(z)$

and

$X_{2}=X\times_{Y}Y_{2}=L_{1}\cup L_{2}=V(x,y)\cup V(x,z).$

We now compute the virtual class of $X$ with respect to the obstruction theory
$E_{X/\mathbb{P}^{3}}=\mathcal{O}_{X}(2)\oplus\mathcal{O}_{X}(2)$. The two
components of $C_{X/C_{Y/\mathbb{P}^{3}}}$ give the following contributions.

  1. (1)

From the first component of $C_{Y/\mathbb{P}^{3}}$ we get a contribution

$C_{X_{1}/Y_{1}}\times_{X_{1}}\mathcal{O}_{Y}(2)\rvert_{X_{1}}=\mathcal{O}_{H}(2)\hookrightarrow\mathcal{O}_{H}(2)\oplus\mathcal{O}_{H}(2)=E_{X/\mathbb{P}^{3}}\rvert_{X_{1}}.$

This gives $2[L]$, where $L$ is the class of a line on the plane $H=V(z)$;

  2. (2)

From the second component of $C_{Y/\mathbb{P}^{3}}$ we get the contribution

$\displaystyle C_{L_{1}\cup
L_{2}/Y_{1}}\times_{X_{2}}\mathcal{O}_{Y}(2)\rvert_{X_{2}}\cong\mathcal{O}_{L_{1}\cup
L_{2}}(2)^{\oplus 2}\hookrightarrow\mathcal{O}_{L_{1}\cup L_{2}}(2)^{\oplus
2}=E_{X/\mathbb{P}^{3}}\rvert_{X_{2}}$

We thus obtain a contribution $[L_{1}]+[L_{2}]$.

In conclusion $[X]^{\mathrm{vir}}=2[L]+[L_{1}]+[L_{2}]$. We can re-write this
as

$[X]^{\mathrm{vir}}=[L_{1}]+3[L].$

######  Example 6.0.4.

Let us now compute the limit $\mathcal{C}_{0}$ defined in Construction 6.0.1.
By Example 3.0.3 we have an embedding

$C_{X/\mathbb{P}^{3}}\hookrightarrow\mathcal{O}_{X}(2)\oplus\mathcal{O}_{X}(2).$

We apply Construction 6.0.1 to the embedding

$\iota_{1}\colon\mathcal{O}_{X}(2)\hookrightarrow\mathcal{O}_{X}(2)\oplus\mathcal{O}_{X}(2).$

Looking at coordinate rings we get that the induced $\iota_{1}$ is

$\displaystyle R[A,B]$ $\displaystyle\twoheadrightarrow R[C]$ $\displaystyle
A$ $\displaystyle\rightarrow C$ $\displaystyle B$ $\displaystyle\rightarrow
0.$

We now deform $\iota_{1}$ to a family of embeddings over $\mathbb{A}^{1}_{t}$.
Consider the morphism in 6.0.1

$\mathcal{O}_{X}(2)\xrightarrow{(\iota_{1},t\cdot
id)}\mathcal{O}_{X}(2)\oplus\mathcal{O}_{X}(2)\oplus\mathcal{O}_{X}(2)\rightarrow\mathcal{O}_{X}(2)\oplus\mathcal{O}_{X}(2).$

For $t\neq 0$, this corresponds to the morphisms of coordinate rings

Let $\mathcal{C}\hookrightarrow\mathcal{E}$ be the flat family in 6.0.1. By
Example 3.0.3 we have that the normal cone $C_{X/\mathbb{P}^{3}}$ is locally
isomorphic to

$\operatorname{Spec}R[A,B]/(yA-xB).$

This shows that $\mathcal{C}\hookrightarrow\mathcal{E}$ is locally given by

$R[A^{\prime},A,B]\twoheadrightarrow
R[A^{\prime},A,B]/(y(A-\frac{1}{t}A^{\prime})-xB).$

Taking the flat limit for $t$ goes to $0$, we get

$\mathcal{C}_{0}\cong R[A^{\prime},B]/yA^{\prime}.$

We see that in this case $\mathcal{C}_{0}=C_{X/C_{Y/Z}}$.

######  Example 6.0.5.

In general, given embeddings $X\hookrightarrow Y\hookrightarrow Z$ and an
embedding of $E_{X/Y}$ into an obstruction bundle $E_{X/Z}$ for
$X\hookrightarrow Z$, the deformation $\mathcal{C}$ of the normal cone inside
the family of cokernels does not necessarily give $\mathcal{C}_{0}\cong
C_{X/C_{Y/Z}}$ (see Proposition 6.0.2 and Example 6.0.4). Consider

$X=V(x,y)\hookrightarrow Y=V(xy)\hookrightarrow Z=\mathbb{P}^{3}_{[x:y:z:w]}.$

We have obstruction theories
$E_{X/Y}=\mathcal{O}_{X}(1)\oplus\mathcal{O}_{X}(1)$ and
$E_{Y/Z}=\mathcal{O}_{Y}(2)$. Set

$E_{X/\mathbb{P}^{3}}=\mathcal{O}_{X}(1)\oplus\mathcal{O}_{X}(1)\oplus\mathcal{O}_{X}(2).$

We now look at the deformation induced by the exact sequence of vector bundles

$0\to\mathcal{O}_{X}(1)\oplus\mathcal{O}_{X}(1)\to\mathcal{O}_{X}(1)\oplus\mathcal{O}_{X}(1)\oplus\mathcal{O}_{X}(2)\to\mathcal{O}_{X}(2)\to
0.$

Let us compute the double deformation space. By restricting to the chart
$\\{w\neq 0\\}$, we see that $\operatorname{Bl}_{Y}Z$ inside
$\operatorname{Bl}_{Y\times\\{0\\}}Z\times\mathbb{A}^{1}_{t}\subseteq\mathbb{A}^{4}_{x,y,z,t}\times\mathbb{P}^{1}_{[\alpha_{0}:\alpha_{1}]}$
is given by $\alpha_{1}=t=0$. This shows that its complement $M^{0}_{Y/Z}$ is
the subscheme $V(xy-t\alpha_{0})\subseteq\mathbb{A}^{5}_{x,y,z,t,\alpha_{0}}$,
and $X\times\mathbb{A}^{1}_{t}$ is $V(x,y,\alpha_{0})$. Similarly, we find
that the double deformation space
$M^{0}_{X\times\mathbb{A}^{1}_{t}/M^{0}_{Y/Z}}$ inside
$\mathbb{A}^{6}_{z,t,s,\beta_{0},\beta_{1}\beta_{2}}$ has equation
$\\{s\beta_{0}\beta_{1}-t\beta_{2}=0\\}$. Here $x=s\beta_{0},\ y=s\beta_{1},\
\alpha_{0}=s\beta_{2}$. Restricting to $s=0$, we see that the flat limit of
$C_{X/Z}\times\\{t\neq 0\\}$ is $C_{X/Z}$. On the other hand, if we restrict
to $t=0$, we find a cone with two components. More precisely, it agrees with
the following

$\displaystyle C_{X/C_{Y/Z}}$
$\displaystyle=C_{X/Y}\times_{X}C_{Y/Z}\rvert_{X}$
$\displaystyle=C_{X/H_{1}}\times_{X}C_{Y/Z}\rvert_{X}\cup
C_{X/H_{2}}\times_{X}C_{Y/Z}\rvert_{X}$
$\displaystyle=\mathcal{O}_{X}(1)\oplus\mathcal{O}_{X}(2)\cup\mathcal{O}_{X}(1)\oplus\mathcal{O}_{X}(2).$

Notice that $C_{X/Z}$ and $C_{X/C_{Y/Z}}$ have different Segre classes, in
particular they cannot lie in a flat family of cones over $X$ (see Proposition
3.4.4). Nevertheless, they are rationally equivalent in the obstruction
bundle, hence the virtual class computation produces the same result

$0^{!}_{E_{X/Z}}([C_{X/Z}])=c_{1}(\mathcal{O}_{X}(2))=2c_{1}(\mathcal{O}_{X}(1))=0^{!}_{E_{X/Y}\oplus
E_{Y/Z}\rvert_{X}}([C_{X/C_{Y/Z}}]).$

######  Remark 6.0.6.

Let us look at $\mathcal{C}_{X\times\mathbb{A}^{1}/M_{Y/Z}^{\circ}}$. In a
first step, we consider
$\operatorname{Bl}_{Y\times\\{0\\}}Z\times\mathbb{A}^{1}_{t}$. Setting $f=xy$,
this is

$\operatorname{Proj}\mathbb{C}[t][x,y,z][\alpha,\beta]/(t\alpha-f\beta).$

We have to remove $\operatorname{Bl}_{Y}Z$ from the central fiber, that is the
locus where $\beta=0$. One sees that $X\times\mathbb{A}^{1}$ is given by
equations $\\{x=y=\alpha=0\\}$. With this, we get that the normal cone
$\mathcal{C}_{X\times\mathbb{A}^{1}/M_{Y/Z}^{\circ}}$ is given by the ring
$\mathbb{C}[t][w_{0},w_{1},w_{2}]/(tw_{2})$. This has two components, one of
them being supported on $\\{t=0\\}$. This shows that
$\mathcal{C}_{X\times\mathbb{A}^{1}/M_{Y/Z}^{\circ}}$ is not flat over
$\mathbb{A}^{1}_{t}$.

##  7\. Properties of virtual classes and computational methods

Virtual classes have good functorial properties which we now list. We then
discuss push-forwards of virtual classes.

###  7.1. Pull-backs of virtual classes.

######  Theorem 7.1.1.

[Man12a] Let $f:X\to Y$ be a morphism of DM type with a perfect obstruction
theory $E^{\bullet}_{X/Y}$. Then $f^{!}_{E_{X/Y}}$ is a bivariant class.

In particular, the following properties hold.

(a) Given a Cartesian diagram

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y}$

with $q$ proper such that $E^{\bullet}_{X/Y}$ is a perfect obstruction theory
for $f$, then for any $\alpha\in A^{*}(Y^{\prime})$, we have that

$p_{*}f^{!}\alpha=f^{!}q_{*}\alpha$

in $A_{*}(Y)$.

(b) Given a Cartesian diagram

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y}$

such that $E^{\bullet}_{X/Y}$ is a perfect obstruction theory for $f$, and
$E^{\bullet}_{Y^{\prime}/Y}$ is a perfect obstruction theory for $g$, then for
any $\alpha\in A^{*}(Y)$, we have that

$g^{!}f^{!}\alpha=f^{!}g^{!}\alpha$

in $A_{*}(X^{\prime})$.

######  Remark 7.1.2.

Let us look again at the line with an embedded point in Example 4.1.1. By
Theorem 7.1.1 and Example 4.0.6, applied to the Cartesian diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathbb{P}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x^{2},xy}$$\textstyle{\mathbb{P}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E}$

we have that

$\displaystyle i_{*}[X]^{\text{vir}}=$ $\displaystyle
c_{\text{top}}(E)\cap[\mathbb{P}^{2}]$ $\displaystyle=$ $\displaystyle 4[pt],$

where $pt$ is a point in $\mathbb{P}^{1}$. By looking at the components of the
normal cone, as we did in Example 4.1.1, we saw that two of those points are
supported on the origin. This is however, irrelevant in this example; since

$A_{0}(X)=A_{0}(X^{\text{red}})=\mathbb{Z},$

we have that all points of $X$ are rationally equivalent.

Example 6.0.3 – a line and a plane – is more meaningful, but has an entirely
analogous flavour. As before, we use Theorem 7.1.1 and Example 4.0.6 to get
that $i_{*}[X]^{vir}=4[H^{2}]$, with $H\in A_{*}(\mathbb{P}^{3})$ the class of
a hyperplane. Although more elaborate, the computation in Example 6.0.3 has
the advantage that it gives a splitting of the virtual class on the components
of $X$.

######  Definition 7.1.3.

Let $X\xrightarrow{f}Y\xrightarrow{g}Z$ be morphisms of DM stacks and let
$E^{\bullet}_{X/Y}$, $E^{\bullet}_{X/Z}$, $E^{\bullet}_{Y/Z}$ be obstruction
theories for $f,g\circ f,g$ respectively. We call the obstruction theories
_compatible_ if there is a morphism of exact triangles:

${f^{*}E^{\bullet}_{Y/Z}}$${E^{\bullet}_{X/Z}}$${E^{\bullet}_{X/Y}}$${f^{*}E^{\bullet}_{Y/Z}[1]}$${f^{*}\mathbb{L}^{\bullet}_{Y/Z}}$${\mathbb{L}^{\bullet}_{X/Z}}$${\mathbb{L}^{\bullet}_{X/Y}}$${f^{*}\mathbb{L}^{\bullet}_{Y/Z}[1]}$$\scriptstyle{g^{\\#}}$$\scriptstyle{f^{\\#}}$

The following is a more general version of Proposition 6.0.2.

######  Theorem 7.1.4.

[Man12a] Let $X\xrightarrow{f}Y\xrightarrow{g}Z$ be morphisms of DM stacks and
$(E^{\bullet}_{X/Y},E^{\bullet}_{X/Z},E^{\bullet}_{Y/Z})$ a triple of
compatible obstruction theories. Then

$(g\circ f)^{!}_{E^{\bullet}_{X/Z}}=f^{!}_{E^{\bullet}_{X/Y}}\circ
g^{!}_{E^{\bullet}_{Y/Z}}.$

###  7.2. Pushforwards of virtual classes.

Recall that, to any proper morphism $f\colon X\to Y$ of DM stacks, we can
associate a pushforward operation

$f_{*}\colon A_{*}(X)\to A_{*}(Y),$

defined on prime cycles by

$f_{*}[V]=\begin{cases}f_{*}[V]=\deg(f|_{V})[f(V)]&\text{if
$\dim(f(V))=\dim(V)$}\\\ 0&\text{otherwise.}\end{cases}$

If $X$ and $Y$ are varieties of the same dimension, then
$f_{*}[X]=\deg(f)[Y]$. If $X$ and $Y$ are pure-dimensional schemes, and
$Y=\bigcup Y_{i}$ with $Y_{i}$ irreducible, then $f_{*}[X]=\sum c_{i}[Y_{i}]$,
for some $c_{i}\in\mathbb{Q}$.

This discussion is generally false for virtual classes. Let us look at an easy
example.

######  Example 7.2.1.

Consider $\tilde{\mathbb{P}}^{2}=\operatorname{Bl}_{p}\mathbb{P}^{2}$ with two
different obstruction theories,

$E_{1}=[\mathcal{O}(-H)\xrightarrow{0}\Omega_{\tilde{\mathbb{P}}^{2}}]\quad\text{and}\quad
E_{2}=[\mathcal{O}(-E)\xrightarrow{0}\Omega_{\tilde{\mathbb{P}}^{2}}]$

with corresponding virtual classes given by
$[\tilde{\mathbb{P}}^{2}]^{\mathrm{vir}}_{1}=H$ and
$[\tilde{\mathbb{P}}^{2}]^{\mathrm{vir}}_{2}=E$. Let
$\operatorname{id}:\tilde{\mathbb{P}}^{2}\to\tilde{\mathbb{P}}^{2}$ denote the
identity. Then, $(\operatorname{id}_{\tilde{\mathbb{P}}^{2}})_{*}[H]\neq[E]$
which shows that

$\operatorname{id}_{*}[\tilde{\mathbb{P}}^{2}]^{\mathrm{vir}}_{1}\neq[\tilde{\mathbb{P}}^{2}]^{\mathrm{vir}}_{2}.$

######  Definition 7.2.2.

We say that a proper morphism of DM type $f\colon X\to Y$ between stacks with
perfect obstruction theories satisfies the _virtual push-forward property_ if

$f_{*}[X]^{\mathrm{vir}}=\sum c_{i}[Y_{i}]^{\mathrm{vir}}$

for some splitting of $[Y]^{\mathrm{vir}}=\sum[Y_{i}]^{\mathrm{vir}}$ and some
$c_{i}\in\mathbb{Q}$.

We say that $f$ satisfies the _strong virtual push-forward_ property if
moreover all $c_{i}$’s are equal.

######  Theorem 7.2.3 ([Man12b]).

Let $f\colon X\to Y$ be a proper morphism of DM type between algebraic stacks.
Assume that $Y$ is connected, and there is a commutative diagram of perfect
obstruction theories

${f^{*}E^{\bullet}_{Y/\operatorname{Spec}(k)}}$${E^{\bullet}_{X/\operatorname{Spec}(k)}}$${f^{*}\mathbb{L}^{\bullet}_{Y/\operatorname{Spec}(k)}}$${\mathbb{L}^{\bullet}_{X/\operatorname{Spec}(k)}}$$\scriptstyle{\phi}$

such that the cone of $\phi$ is supported in degrees $[-1,0]$. Then $f$
satisfies the strong virtual push-forward property in homology.

In practice, the cone of $\phi$ is supported in degrees $[-1,0]$ iff

$h^{-1}(\phi)\colon h^{-1}(f^{*}E^{\bullet}_{Y/\operatorname{Spec}(k)})\to
h^{-1}(E^{\bullet}_{X/\operatorname{Spec}(k)})$

is injective, or dually iff the map

$h^{1}(\phi^{\vee})\colon h^{1}(E^{\bullet\vee}_{X/\operatorname{Spec}(k)})\to
h^{1}(f^{*}E^{\bullet\vee}_{Y/\operatorname{Spec}(k)})$

is surjective.

The condition in Theorem 7.2.3 is not necessary.

######  Example 7.2.4.

Let $X$ be the union of two smooth divisors $D_{1}$, $D_{2}$ on a smooth space
$V$. Then the obvious virtual class is

$[X]^{\mathrm{vir}}=[D_{1}]+[D_{2}]$

and we see that the embedding $i_{1}:D_{1}\to X$ satisfies the virtual push-
forward property but we do not necessarily have a surjective morphism between
obstruction theories. We do have a morphism between obstruction theories
$O_{D_{1}}(D_{1})\to O_{D_{1}}(D_{1}+D_{2})$ given by multiplication with
$D_{2}$. This is an isomorphism if $D_{1}$ does not intersect $D_{2}$, but it
is not surjective if the intersection is non-empty.

Let us give a new virtual push-forward Theorem which is better than 7.2.3 in
the sense that it covers the example 7.2.4, but has the disadvantage of a
hypothesis which may be hard to check in general.

######  Theorem 7.2.5.

Let $X\stackrel{{\scriptstyle f}}{{\hookrightarrow}}Y\hookrightarrow V$ be
embeddings of DM stacks such that $X$ and $Y$ have obstruction bundles
$E_{X/V}$ and $E_{Y/V}$ of the same rank. Consider the complexes
${E^{\bullet}}_{Y/V}=[E^{\vee}_{Y/V}\to 0]$ and
$E^{\vee}_{X/V}=[E^{\vee}_{X/V}\to 0]$ supported in $[-1,0]$ and ssume that
there is a commutative diagram of perfect obstruction theories

${f^{*}{E^{\bullet}}_{Y/V}}$${{E^{\bullet}}_{X/V}}$${f^{*}\mathbb{L}^{\bullet}_{Y/V}}$${\mathbb{L}^{\bullet}_{X/V}.}$$\scriptstyle{\phi^{\vee}}$

Let $f^{*}C_{Y/V}$ denote the fibre product $C_{Y/V}\times_{Y}X$, let

$C=f^{*}C_{Y/V}\times_{f^{*}E_{Y/V}}E_{X/V}$

and consider the induced diagram

${C_{X/V}}$${C}$${f^{*}C_{Y/V}}$${E_{X/V}}$${f^{*}E_{Y/V}.}$$\scriptstyle{j}$$\scriptstyle{\phi}$

  1. (1)

The complex

$F^{\bullet}=[E_{X/V}\to f^{*}E_{Y/V}]$

concentrated in $[0,1]$ is a perfect dual obstruction theory for $\phi$.

  2. (2)

If $j_{*}[C_{X/V}]=N\phi_{F}^{!}[f^{*}C_{Y/V}]$ for some $N\in\mathbb{Q}$,
then $f$ satisfies the virtual push-forward property.

###### Proof.

The first claim follows from the triangle of cotangent complexes associated to
$E_{X/V}\to f^{*}E_{Y/V}\to X$.

For the second one denote
$[Y]^{\mathrm{vir}}|_{X}=0_{f^{*}E_{Y/V}}^{!}[f^{*}C_{Y/V}]$. Then, by
functoriality of virtual pull-backs, we have that

$[Y]^{\mathrm{vir}}|_{X}=0^{!}_{E_{X/V}}\phi_{F}^{!}[f^{*}C_{Y/V}]$

By hypothesis we have $j_{*}[C_{X/V}]=N\phi_{F}^{!}[f^{*}C_{Y/V}]$. This shows
that

(12) $[Y]^{\mathrm{vir}}|_{X}=\frac{1}{N}[X]^{\mathrm{vir}}.$

In the following we prove that

(13) $f_{*}[Y]^{\mathrm{vir}}|_{X}=\sum c_{i}[G_{i}],$

for $G_{i}$ such that $[Y]^{\mathrm{vir}}=\sum[G_{i}]$. By commutativity of
virtual pullbacks with push-forwards in the following Cartesian diagram

${X}$${Y}$${f^{*}E_{Y/V}}$${E_{Y/V}}$

it is enough to check that $\phi_{*}[f^{*}C_{Y/V}]=\sum d_{i}[\tilde{G}_{i}]$,
with $\tilde{G}_{i}$ the irreducible components of $C_{Y/V}$. This is now
clear because $f^{*}C_{Y/V}$ and $C_{Y/V}$ are both pure of the same
dimension.

The conclusion follows from (12) and (13). ∎

######  Remark 7.2.6.

Note that $C_{Y/V}\times_{Y}X$ could have components of dimension smaller than
the dimension of $V$. In this case $j_{*}[C_{X/V}]\neq
N\phi_{F}^{!}[f^{*}C_{Y/V}]$. The example of $f$ we have in mind is an
embedding of components. Note that in the above proof $c_{i}$ can be zero.

###  7.3. Computations of virtual classes

####  7.3.1. Computations of virtual classes via Segre classes

Given a global embedding $i:X\to V$ and an obstruction theory $E_{X/V}$, the
computation of the virtual class reduces to Segre classes computations (see
Proposition 4.0.5). Segre classes have been computed in many instances (see
e.g. [Alu03, AB03, Alu16, Hel16, Hel17, EJP13, MQ13]). Although existent
computations are usually for $i_{*}s(X,V)$, ideas can probably be adapted to
compute $[X]^{\mathrm{vir}}$ instead of $i_{*}[X]^{\mathrm{vir}}$.

####  7.3.2. Method 1

Consider the blow-up diagram

${\tilde{X}}$${\tilde{Y}}$${X}$${Y}$$\scriptstyle{\eta}$

Then, given an obstruction bundle $E_{X/Y}$, $\eta^{*}E_{X/Y}$ is an
obstruction bundle for $\tilde{X}$ in $\tilde{Y}$. There is an exact sequence
of vector bundles

$0\to\mathcal{O}_{\tilde{X}}(\tilde{X})\to\eta^{*}E_{X/Y}\to Q\to 0$

and

$[X]^{\mathrm{vir}}=\eta_{*}c_{\text{top}}(Q)\cap[\tilde{X}]$

This method is used in Examples 3.5.3 and 4.1.2.

######  Remark 7.3.1.

This is the basic idea of Aluffi [Alu03]. Although simple enough, it might be
complicated to implement it when $X$ and $V$ are moduli spaces. Blowing them
up might mean loosing modularity, and then Chern classes computations might
become more difficult.

######  Remark 7.3.2.

Other ideas have been used successfully to compute Segre classes. For example
[EJP13] use residual schemes. As far as we know, these ideas have not been
applied to our context yet.

####  7.3.3. Computations of virtual classes via deformation

####  7.3.4. Method 2

Given an embedding $X\hookrightarrow V$ and a set of global generators
$\\{f_{1},...,f_{k}\\}$ for the ideal $I_{X/V}$, we consider the obstruction
bundle defined by restricting $\oplus\mathcal{O}_{V}(f_{i})$ to $X$. Set
$Z_{k+1}=V$ and $i_{j}\colon Z_{j}=V(f_{j})\cap Z_{j+1}\hookrightarrow
Z_{j+1}$. Note that $Z_{j}$ is a union of divisors and components in
$Z_{j+1}$. We use this sequence of embeddings to deform $C_{Z_{j}/V}$ to
$C_{Z_{j}/C_{Z_{j+1}/V}}$. The virtual class on $X$ is then
$[X]^{\mathrm{vir}}=i_{0}^{!}(i_{1}^{!}(...i_{k}^{!}[V]))$. Each pull-back can
be computed as in Example 6.0.3. This is easy, but very restrictive: it
applies only to (possibly non-transverse) intersections of divisors.

This method is used in Examples 6.0.3 and 6.0.5. It could also be applied to
compute the virtual class in Example 4.1.1.

####  7.3.5. Generalization

More generally, assume that the obstruction theory of $X$ in $V$ is the
restriction of a vector bundle $E$ on $V$ with a surjection $E^{\vee}\to
I_{X/V}$. Assume furthermore that $E$ admits a filtration $0\subseteq
E_{1}\subseteq\ldots\subseteq E_{r}=E$, with line bundle quotients
$L_{k}=E_{k}/E_{k-1}$. By dualising the maps induced by the filtration we get
diagrams:

${L_{k}^{\vee}}$${E_{k}^{\vee}}$${E_{k-1}^{\vee}}$${i_{k}^{*}I_{X/V}}$${\mathcal{O}_{V_{k}}}$

hence (possibly vanishing) cosections $f_{k}\colon
L_{k}^{\vee}\to\mathcal{O}_{V_{k}}$, where $i_{k}\colon V_{k}\hookrightarrow
V$ is defined iteratively by the vanishing of $\\{f_{r},\ldots,f_{k+1}\\}$.
Note that by this definition we have maps $E_{k-1}^{\vee}\to
i_{k-1}^{*}I_{X/V}$. From Construction 6.0.1 and proposition 6.0.2 we get an
embedding of a cone inside a split vector bundle

$C_{X/C_{V_{1}/\ldots_{/C_{V_{r-1}/V}}}}\subseteq\bigoplus_{k=1}^{r}L_{k}.$

##  8\. Applications to moduli spaces of stable maps

In this section we illustrate the theory in the previous sections in more
meaningful examples. We define moduli spaces of stable maps and look at their
irreducible components. We define Gromov–Witten invariants and we discuss the
quantum Lefschetz hyperplane property in genus zero. In genus one the quantum
Lefschetz hyperplane property fails. One way to fix this problem is by first
separating degenerate contributions. We show how to do this in one simple
example.

###  8.1. Moduli spaces of stable maps and Gromov–Witten invariants

######  Definition 8.1.1.

Let $X$ be a smooth projective variety over $\mathbb{C}$, $\beta\in
H_{2}(X,\mathbb{Z})^{+}$, $(g,n)\in\mathbb{N}^{2}$. The moduli space of stable
maps $\overline{\mathcal{M}}_{g,n}(X,\beta)$ is the set of isomorphism classes
of maps $[f:(C,x_{1},\ldots,x_{n})\to X]$ such that:

  1. (1)

$C$ is a projective, connected, reduced, nodal curve of genus $g$,

  2. (2)

$x_{1},\ldots,x_{n}$ are distinct smooth points of $C$,

  3. (3)

the map $f$ satisfies $f_{*}[C]=\beta$,

  4. (4)

stability condition: any contracted component of geometric genus 0 has at
least 3 special points (i.e. nodes or markings), and any contracted curve of
genus 1 has at least 1 special point.

######  Theorem 8.1.2.

$\overline{\mathcal{M}}_{g,n}(X,\beta)$ is a proper DM stack [Kon95]. Let $X$
be a homogeneous space, e.g. $X=\mathbb{P}^{N}$. The stack
$\overline{\mathcal{M}}_{0,n}(X,\beta)$ is smooth and irreducible [FP97].

####  8.1.1. Obstruction theory from deformation theory

Let $\mathfrak{M}_{g,n}$ be the algebraic stack whose points parametrise pre-
stable genus $g$ curves with $n$ marked points. A deformation theory argument
shows that $\mathfrak{M}_{g,n}$ is a smooth Artin stack of dimension $3g-3+n$.
Let $\rho\colon\overline{\mathcal{M}}_{g,n}(X,\beta)\to\mathfrak{M}_{g,n}$
denote the morphism which forgets the map. In the following we introduce an
obstruction theory for $\rho$. Consider the morphisms

---
$\textstyle{\mathcal{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\textstyle{X}$$\textstyle{\overline{\mathcal{M}}_{g,n}(X,\beta)}$

where $\pi:\mathcal{C}\to\overline{\mathcal{M}}_{g,n}(X,\beta)$ is the
universal curve and $f:C\to X$ is the universal map. A relative dual
obstruction theory for $\rho$ is given by the complex

(14) $\mathfrak{E}^{\bullet}_{\rho}=R^{\bullet}\pi_{*}f^{*}T_{X}.$

We define the _expected dimension_ of $\overline{\mathcal{M}}_{g,n}(X,\beta)$
as

$\operatorname{expdim}(\overline{\mathcal{M}}_{g,n}(X,\beta))=(1-g)(\dim(X)-3)-K_{X}\cdot\beta+n.$

####  8.1.2. Gromov–Witten invariants

The obstruction theory in (14) induces a virtual class

$[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}_{\mathfrak{E}^{\bullet}_{\rho}}\in\;\;A_{\mathrm{expdim}}(\overline{\mathcal{M}}_{g,n}(X,\beta)).$

Consider the evaluation maps

$\mathrm{ev}_{i}\colon\overline{\mathcal{M}}_{g,n}(X,\beta)\to X$

defined by $\mathrm{ev}_{i}(C,x_{1},...,x_{n},f)=f(x_{i}).$ Consider
$\gamma_{i}\in A^{k_{i}}(X)$ such that

$\sum_{i}{k_{i}}=\mathrm{expdim}(\overline{\mathcal{M}}_{g,n}(X,\beta)).$

We call

$\mathrm{deg}\left([\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}_{\mathfrak{E}^{\bullet}_{\rho}}\cap(\mathrm{ev}_{1}^{*}\gamma_{1}\cup\dots\cup\mathrm{ev}_{n}^{*}\gamma_{n})\right)\in\mathbb{Q}$

a Gromov-Witten invariant with insertions $\gamma_{i}$.

###  8.2. Examples of Moduli spaces of stable maps

######  Example 8.2.1.

We have an isomorphism
$\overline{\mathcal{M}}_{g,n}(X,0)\simeq\overline{\mathcal{M}}_{g,n}\times X$,
which shows that $\overline{\mathcal{M}}_{g,n}(X,0)$ has dimension
$3g-3+n+\dim(X)$. Note that this is bigger than the expected dimension. The
difference is $g\dim(X)$, which is the rank of the obstruction bundle
$\mathbb{E}^{\vee}\boxtimes T_{X}$. Here, $\mathbb{E}$ denotes the Hodge
bundle $\pi_{*}\omega_{\pi}$. The fibres of the obstruction bundle are

$H^{1}(C,f^{*}T_{X})=H^{1}(C,\mathcal{O}_{C})\otimes T_{X}.$

######  Example 8.2.2.

Let us look at $\overline{\mathcal{M}}_{1,0}(\mathbb{P}^{2},3)$ and show it
has several irreducible components.

  * •

The _main_ component is the closure of the open locus where the source curve
is smooth irreducible and the map is an embedding. Recall that a smooth cubic
has genus one by the degree-genus formula. The main component has the same
dimension as the space of cubics in $\mathbb{P}^{2}$, that is 9.

  * •

$D^{1}$ is the closure of the locus where the source curve has two components:
one rational curve glued to an elliptic curve and the map contracts the
elliptic curve and sends the rational curve to a nodal cubic. To compute the
dimension of this locus we have to choose a nodal cubic, a point on it
corresponding to the image of the genus one curve, and an elliptic curve with
one marked point. Adding these dimensions we get

$10=8+1+1.$

  * •

$D^{2}$ is the closure of the locus where the source curve consists of two
rational tails mapping to a line and a conic respectively, and an elliptic
bridge which gets contracted. To compute the dimension of this locus we have
to choose a conic, a line in the plane, and a point of
$\overline{\mathcal{M}}_{1,2}$. Adding these up, we get

$9=5+2+2.$

######  Example 8.2.3.

More generally, $\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)$ has several
irreducible components: a main component
$\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)^{\mathrm{main}}$, which is the
closure of the locus of maps from a smooth elliptic curve, and boundary
components

$D(\mathbb{P}^{r},d)^{\lambda}=\overline{\mathcal{M}}_{1,k+n_{0}}(\mathbb{P}^{r},0)\times_{\mathbb{P}^{r}}\overline{\mathcal{M}}_{0,n_{1}}(\mathbb{P}^{r},d_{1})\times_{\mathbb{P}^{r}}\cdots\times_{\mathbb{P}^{r}}\overline{\mathcal{M}}_{0,n_{k}}(\mathbb{P}^{r},d_{k}).$

Here, $\lambda$ denotes the data $(k,d_{1},...d_{k},n_{0},...,n_{k})$ such
that $\sum d_{i}=d$ and $\sum n_{i}=n$. We have that
$D(\mathbb{P}^{r},d)^{\lambda}$ are either components or closed substacks of
the main component (if $k>r$). In the following we denote by
$D^{1}(\mathbb{P}^{r})$ the boundary component $D(\mathbb{P}^{r},d)^{\lambda}$
with $\lambda=(1,d,0,n)$. With this notation we have that

$\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)=\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)^{\mathrm{main}}\bigcup_{\lambda}D(\mathbb{P}^{r},d)^{\lambda}.$

######  Example 8.2.4.

$\overline{\mathcal{M}}_{g,0}(\mathbb{P}^{1},1)$ has an empty “main component”
for $g>0$, since there is no map of degree 1 from a smooth curve of positive
genus to $\mathbb{P}^{1}$. Yet, this space is not empty – it contains maps
whose source curve is a smooth rational curve $R$ that is mapped
isomorphically to the target glued to $l$ contracted branches of positive
genus $g_{1},\ldots,g_{l}$ such that $\sum_{i=1}^{l}g_{i}=g$. Note that the
restriction of the map to $R$ is an embedding and thus the dual graph of the
source curve must be a tree around $R$. The dimension of such loci is
$l+\sum(3g_{i}-3+1)=3g-l$.

######  Example 8.2.5.

Let $\tilde{\mathbb{P}}^{2}$ be the blow-up of the projective plane at a point
$p$. Let $H$ denote the class of the the pullback of a general hyperplane
class, and $E$ the class of the exceptional divisor. Then
$H_{2}(\tilde{\mathbb{P}}^{2},\mathbb{Z})=\mathbb{Z}\langle H,E\rangle$ and
the effective cone is $\mathbb{N}\langle H-E,E\rangle$.

Consider $\overline{\mathcal{M}}_{0,0}(\tilde{\mathbb{P}}^{2},3H)$. This space
has several components as follows.

  * •

One component of dimension 8, whose general point consists of a rational plane
(nodal) cubic that does not pass through $p$.

  * •

Another component of dimension 8 isomorphic to

$\overline{\mathcal{M}}_{0,1}(\tilde{\mathbb{P}}^{2},3H-2E)\times_{\tilde{\mathbb{P}}^{2}}\overline{\mathcal{M}}_{0,1}(\tilde{\mathbb{P}}^{2},2E).$

The general point consists of maps from a curve with two components, one
mapping in $\mathbb{P}^{2}$ to a cubic with a node in $p$, and the other one
covering the exceptional divisor 2:1. Note that
$\overline{\mathcal{M}}_{0,1}(\tilde{\mathbb{P}}^{2},2E)\simeq\overline{\mathcal{M}}_{0,1}(\mathbb{P}^{1},2)$
because $E$ is rigid. $\overline{\mathcal{M}}_{0,1}(\mathbb{P}^{1},2)$ has
dimension 3, but here it comes with a non-trivial obstruction bundle.

######  Remark 8.2.6.

We conclude this subsection with the following remarks.

  * •

For general $g$ and target $X$, the space of stable maps may have many
components of different dimensions.

  * •

It is difficult to understand what lies in the boundary of the main component
and consequently the tangent space (or the intrinsic normal cone) at those
points.

###  8.3. Equations for the moduli space of maps to $\mathbb{P}^{r}$.

######  Notation 8.3.1.

Consider the Artin stack $\mathfrak{M}^{div}_{1,n}$ parametrising genus 1 pre-
stable curves with $n$ marked points and a Cartier divisor $D$ of degree $d$
contained in the smooth locus of $C$. We denote by
$\mathfrak{C}_{\mathfrak{M}^{div}}$ the universal curve.

We want to find a local embedding of
$\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)$ into a smooth stack $V$, and
equations for $\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)\hookrightarrow
V$. This was done in [CFK10], [HL10]. We sketch the construction and refer to
[HL10] for details.

The moduli space $\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)$ can be
constructed as an open subset of the moduli space that parametrises

$(C;x_{1},\ldots,x_{n};L;s_{0},\ldots,s_{r}),$

with $L$ a line bundle on $C$ and $s_{i}\in H^{0}(C,L)$. On the open locus
$U_{0}\subseteq\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)$ where
$D=\\{s_{0}=0\\}$ does not contain nodes or components, we have a projection
$U_{0}\to\mathfrak{M}^{div}_{1,n}$. On a chart $\mathcal{V}$ of
$\mathfrak{M}^{div}_{1,n}$ we may pick a section of the universal curve
$\mathfrak{C}_{\mathfrak{M}^{div}}$ that passes through the smooth locus of
the minimal subcurve of genus 1. Denote by $\mathcal{A}$ the corresponding
relative Cartier divisor. Let $\mathcal{L}$ be the universal line bundle on
$\mathfrak{C}_{\mathfrak{M}^{div}}$ and consider the exact sequence on
$\mathfrak{C}_{\mathcal{V}}:$

$0\to\mathcal{L}\to\mathcal{L}(\mathcal{A})\to\mathcal{L}(\mathcal{A})|_{\mathcal{A}}\to
0$

By pushing it forward along the universal curve it gives us an exact sequence
of sheaves on $\mathcal{V}$

(15)
$0\to\pi_{*}\mathcal{L}\to\pi_{*}\mathcal{L}(\mathcal{A})\to\pi_{*}\mathcal{L}(\mathcal{A})|_{\mathcal{A}}\to
R^{1}\pi_{*}\mathcal{L}\to 0.$

Then $\pi_{*}\mathcal{L}(\mathcal{A})$ is locally free on $\mathcal{V}$, and
we define

$V=\bigoplus_{1}^{r}\pi_{*}\mathcal{L}(\mathcal{A}).$

The exact sequence above can be pulled back to $V$ and we get a tautological
section of $\pi_{*}\mathcal{L}(\mathcal{A})|_{\mathcal{A}}^{\oplus r}$, whose
vanishing locus contains the local chart $U_{0}$ of
$\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)$ as an open substack.

If we look around a point of the form: a contracted elliptic curve $E$ glued
to one rational tail $R$, it turns out that there are local coordinates
$y_{i},x_{1,i},\ldots,x_{d,i},\ i=1,\ldots,r$ on $V$ such that the equations
of $U_{0}\subseteq\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)$ inside $V$
are of the form

(16) $\\{\xi y_{1}=\ldots=\xi y_{r}=0\\}$

where $\xi$ is a local coordinate on $\mathcal{V}$ which vanishes on the locus
where the node separating $E$ from $R$ is not smoothed out.

Around a point of the form: contracted elliptic curve with $k$ rational tails
of positive degree, equations look more complicated, but after a suitable
blow-up, they can be brought to the form above. See [HL10] for a complete
discussion.

######  Remark 8.3.2.

The idea of twisting a line bundle on a family of curves with a sufficiently
high power of a Cartier divisor appears in [CFK10, CFKM14]. In this way one
gets a global embedding, but equations for this embedding are not easy to
find.

By twisting with just one appropriately chosen section one can get a smaller
smooth ambient space, together with very explicit equations. This is the
approach in [HL10, Thm. 4.16]. The drawback is that this construction is only
local on $\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)$.

###  8.4. Obstruction theory from local equations

Note that the embedding $U_{0}\hookrightarrow V$, endows $U_{0}$ with a
perfect obstruction theory relative to $\mathfrak{M}^{div}_{1,n}$

$[\pi_{*}\mathcal{L}(\mathcal{A})\to\pi_{*}\mathcal{L}(\mathcal{A})|_{\mathcal{A}}].$

We give explicit gluing maps for the above local obstruction theories. We view
this construction as an alternative definition of the obstruction theory for
$\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)$.

Fix $\mathcal{A}_{1}$, $\mathcal{A}_{2}$, $\mathcal{A}_{3}$ and
$\tilde{\mathcal{A}}$ large enough such that
$R^{1}\pi_{*}\mathcal{L}(\mathcal{A}_{i})=0$ and such that
$\tilde{\mathcal{A}}-\mathcal{A}_{i}$ is effective for every $i$. Denote by
$A_{i}^{\bullet}$ the complex
$[\pi_{*}\mathcal{L}(\mathcal{A}_{i})\to\pi_{*}\mathcal{L}(\mathcal{A}_{i})|_{\mathcal{A}_{i}}]$
concentrated in $[0,1]$ and by $\tilde{A}^{\bullet}$ the complex
$[\pi_{*}\mathcal{L}(\tilde{\mathcal{A}})\to\pi_{*}\mathcal{L}(\tilde{\mathcal{A}})|_{\tilde{\mathcal{A}}}]$
concentrated in $[0,1]$. Consider commutative diagrams

$\textstyle{\mathcal{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}(\mathcal{A}_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}(\tilde{\mathcal{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}(\mathcal{A}_{i})|_{\mathcal{A}_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}(\tilde{\mathcal{A}})|_{\tilde{\mathcal{A}}}.}$

Take push-forwards via $\pi$ of the morphisms above. The Snake Lemma, applied
to the induced diagram, implies that $A_{i}^{\bullet}\to\tilde{A}^{\bullet}$
are quasi-isomorphisms. These quasi-isomorphisms induce quasi-isomorphisms
$f_{ij}:A_{i}^{\bullet}\to A_{j}^{\bullet}$ via the roof

$\textstyle{A_{j}^{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{i}^{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{A}^{\bullet}}$$\textstyle{A_{j}^{\bullet}}$

and it is easy to see that these quasi-isomorphisms satisfy the cocycle
condition. Then $\bigoplus^{r}A_{i}^{\bullet}$ together with the gluing
morphisms $f_{ij}$ define a perfect obstruction theory for
$\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)$ relative to
$\mathfrak{M}^{div}_{1,n}.$

###  8.5. Genus 0 invariants of a hypersurface in projective space.

Let $s\in H^{0}(\mathbb{P}^{r},\mathcal{O}_{\mathbb{P}^{r}}(a))$ be a generic
homogeneous polynomial of degree $a$, and let $X=Z(s)$ be the hypersurface in
$\mathbb{P}^{r}$ cut out by $s$. Note that the inclusion $i\colon
X\hookrightarrow\mathbb{P}^{r}$ induces a morphism
$j\colon\overline{\mathcal{M}}_{g,n}(X,\beta)\hookrightarrow\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},i_{*}\beta)$.
We are interested in computing
$j_{*}[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\text{vir}}$. It turns out that
the answer is simple when $g=0$ and particularly hard in higher genus.

When $g=0$, $\mathcal{E}=\pi_{*}f^{*}\mathcal{O}_{\mathbb{P}^{r}}(a)$ is a
vector bundle on $\overline{\mathcal{M}}_{0,n}(\mathbb{P}^{r},d)$.
Furthermore, $s$ induces a section $\eta$ of $\mathcal{E}$. In each point
$(C,u_{0},\ldots,u_{r})$, with $(u_{0},\ldots,u_{r})\in
H^{0}(C,f^{*}\mathcal{O}(1))$, $\eta$ is $s(u_{0},\ldots,u_{r})$. This shows
that $\eta$ vanishes precisely on the locus of stable maps to $\mathbb{P}^{r}$
whose image lies inside $X$.

######  Theorem 8.5.1 ([KKP03]).

Let $\overline{\mathcal{M}}_{0,n}(X,d)=\bigsqcup_{\\{\beta\colon
i_{*}\beta=d\\}}\overline{\mathcal{M}}_{0,n}(X,\beta)$. Then,

$j_{*}[\overline{\mathcal{M}}_{0,n}(X,d)]^{\mathrm{vir}}=c_{ad+1}(\mathcal{E})\cap[\overline{\mathcal{M}}_{0,n}(\mathbb{P}^{r},d)].$

This can be seen as a functoriality statement in intersection theory in the
following way. We have a commutative diagram

${\overline{\mathcal{M}}_{0,n}(X,d)}$${\overline{\mathcal{M}}_{0,n}(\mathbb{P}^{r},d)}$${\mathfrak{M}_{0,n}}$$\scriptstyle{j}$$\scriptstyle{\rho_{X}}$$\scriptstyle{\rho_{\mathbb{P}}}$

equipped with a compatible triple of dual obstruction theories as follows. The
exact sequence of vector bundles on $X$

$0\to T_{X}\to T_{\mathbb{P}^{r}|X}\to
N_{X/\mathbb{P}^{r}}\simeq\mathcal{O}_{X}(a)\to 0$

induces the following exact triangle on $\overline{\mathcal{M}}_{0,n}(X,d)$

(17)
${R^{\bullet}\pi_{*}f^{*}(T_{X})}$${R^{\bullet}\pi_{*}f^{*}(T_{\mathbb{P}^{r}|X})}$${R^{\bullet}\pi_{*}f^{*}(N_{X/\mathbb{P}^{r}})}$${R^{\bullet}\pi_{*}f^{*}(T_{X})[1]}$${\mathbb{T}_{\overline{\mathcal{M}}(X)/\mathfrak{M}}}$${j^{*}\mathbb{T}_{\overline{\mathcal{M}}(\mathbb{P}^{r})/\mathfrak{M}}}$${\mathbb{T}_{\overline{\mathcal{M}}(X)/\overline{\mathcal{M}}(\mathbb{P}^{r})}[1]}$${\mathbb{T}_{\overline{\mathcal{M}}(X)/\mathfrak{M}}[1].}$

Here we have denoted $\overline{\mathcal{M}}_{0,n}(X,d)$ by
$\overline{\mathcal{M}}(X)$, $\overline{\mathcal{M}}_{0,n}(\mathbb{P}^{r},d)$
by $\overline{\mathcal{M}}(\mathbb{P}^{r})$ and , and $\mathfrak{M}_{0,n}$ by
$\mathfrak{M}$. Moreover, we have

  1. (1)

$R^{\bullet}\pi_{*}f^{*}(T_{\mathbb{P}^{r}|X})\simeq
j^{*}R^{\bullet}\pi^{\mathbb{P}}_{*}f_{\mathbb{P}}^{*}(T_{\mathbb{P}^{r}})$
(it follows from flat base-change in the derived category); this complex is
supported in degree 0.

  2. (2)

$R^{\bullet}\pi_{*}f^{*}(N_{X/\mathbb{P}^{r}})\simeq\mathcal{E}$ is a vector
bundle, supported in degree 0, and it gives a dual perfect obstruction theory
for $j$.

Let $\rho_{\mathbb{P}}^{!}$ and $\rho_{X}^{!}$ be the virtual pullbacks with
respect to the compatible obstruction theories in (17), and let $j^{!}$ be the
Gysin pullback induced by the section $\eta\in
H^{0}(\overline{\mathcal{M}}_{0,n}(\mathbb{P}^{r},d),\mathcal{E})$. By (1) we
have

$[\overline{\mathcal{M}}_{0,n}(\mathbb{P}^{r},d)]^{\mathrm{vir}}=[\overline{\mathcal{M}}_{0,n}(\mathbb{P}^{r},d)].$

By functoriality of virtual pullbacks [Man12a] we have

$[\overline{\mathcal{M}}_{0,n}(X,d)]^{\text{vir}}=\rho_{X}^{!}[\mathfrak{M}_{0,n}]=j^{!}\rho_{\mathbb{P}}^{!}[\mathfrak{M}_{0,n}]=j^{!}[\overline{\mathcal{M}}_{0,n}(\mathbb{P}^{r},d)].$

By Theorem 7.1.1 and Example 4.0.6 applied to the Cartesian diagram

$\textstyle{\overline{\mathcal{M}}_{0,n}(X,d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{\overline{\mathcal{M}}_{0,n}(\mathbb{P},d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{\overline{\mathcal{M}}_{0,n}(\mathbb{P},d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\textstyle{\mathcal{E}}$

we get
$j_{*}[\overline{\mathcal{M}}_{0,n}(X,d)]^{\mathrm{vir}}=c_{ad+1}(\mathcal{E})\cap[\overline{\mathcal{M}}_{0,n}(\mathbb{P}^{r},d)]$.

###  8.6. Genus 1 invariants of a hypersurface in projective space.

The above story fails in higher genus. From now on we look at $g=1$. Unlike
for $g=0$, the sheaf $\mathcal{E}=\pi_{*}f^{*}\mathcal{O}_{\mathbb{P}^{r}}(a)$
is not a vector bundle. $\mathcal{E}$ has rank $ad$ on the open consisting of
maps with smooth domain and it has rank $ad+1$ on
$D^{\lambda}(\mathbb{P}^{r})$. To see this, we look at
$H^{1}(C,f^{*}T_{\mathbb{P}^{r}})$. On a smooth genus one curve, this group is
zero. If $E$ is a contracted curve of genus one, then
$f^{*}\mathcal{O}(a)|_{E}\simeq\mathcal{O}_{E}$ is trivial and
$H^{1}(E,\mathcal{O}_{E})\simeq\mathbb{C}$. In terms of obstruction theories,
this means that the dual relative obstruction theory of

$j\colon\overline{\mathcal{M}}_{1,n}(X,d)\to\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)$

is not supported in degree $1$, but in degrees $[1,2]$.

####  8.6.1. The Vakil–Zinger desingularisation

A first approach to define invariants with a better enumerative behaviour was
proposed by Zinger, Li–Zinger and Vakil–Zinger [Zin09a], [VZ08], [LZ09a],
[LZ09b], [Zin08], [Zin09b]. The rough idea is to blow-up the boundary
components until we find a desingularisation of the main component. Moreover,
the object defined analogously to $\mathcal{E}$ on this desingularisation is a
vector bundle. For this we first introduce an additional stack of decorated
pre-stable curves.

######  Construction 8.6.1.

Let us define the moduli space of weighted pre-stable curves
$\mathfrak{M}_{1,n}^{\text{wt}}$: this is an Artin stack which parametrises
the data $(C,x_{1},...,x_{n},\underline{d})$, where $C$ is a prestable curve,
$x_{1},...,x_{n}$ markings, and if $C$ has $j$ irreducible components
$\underline{d}=(d_{1},...,d_{j})\in\mathbb{Z}^{j}$. Then
$\mathfrak{M}_{1,n}^{\text{wt}}$ is étale (but not separated) over
$\mathfrak{M}_{1,n}$.

Again, for any partition $\lambda$ of $(d=\sum\underline{d},n)$, we can define
$\mathfrak{D}^{\lambda}$ the closed substack of
$\mathfrak{M}_{1,n}^{\text{wt}}$ the general point of which is a smooth
elliptic curve of weight 0, with $k$ rational tails attached to it, with
weight and markings distributed according to $\lambda$. In fact, this is finer
than necessary, and we may as well group all the $\lambda$ with the same
number of positive rational tails $k$ together into $\mathfrak{D}^{k}$. Note
that these are closed substacks of $\mathfrak{M}_{1,n}^{\text{wt}}$ of
codimension $k$.

We now briefly describe the Vakil-Zinger desingularisation following [VZ08]
and [HL10]. The idea is to start with the moduli space of pre-stable weighted
curves $\mathfrak{M}_{1,n}^{\text{wt}}$ and construct an Artin stack
$\widetilde{\mathfrak{M}}_{1,n}^{\text{wt}}$ by an iterated blow-up. We first
blow up $\mathfrak{D}^{1}$ inside $\mathfrak{M}_{1,n}^{\text{wt}}$ (this has
no effect), then the strict transform of $\mathfrak{D}^{2}$, which is smooth,
and so on. Note that the total weight is constant on connected components of
$\mathfrak{M}_{1,n}^{\text{wt}}$, and for a fixed weight it is necessary to
perform only finitely many blow-up steps. The Vakil-Zinger space is then
constructed by taking the following Cartesian diagram

${\widetilde{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)}$${\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)}$${f\colon(C,x)\to\mathbb{P}^{r}}$${\widetilde{\mathfrak{M}}_{1,n}^{\text{wt}}}$${\mathfrak{M}_{1,n}^{\text{wt}}}$${(C,x,\mathrm{deg}(f^{*}\mathcal{O}(1)).}$

Here the right vertical map is obtained by forgetting the map, but remembering
the degree of $f^{*}\mathcal{O}_{\mathbb{P}^{r}}(1)$ on the irreducible
components of $C$.

######  Theorem 8.6.2.

(i) The blow-up of the main component
$\widetilde{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)^{\mathrm{main}}$ is smooth.

(ii) Let
$\pi^{\circ}:\mathcal{C}^{\mathrm{main}}\to\widetilde{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)^{\mathrm{main}}$
be the universal curve over the main component of
$\widetilde{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)$ and let
$f^{\circ}:\mathcal{C}^{\mathrm{main}}\to\mathbb{P}^{r}$ be the universal
morphism. Then
$\mathcal{E}_{ad}:=\pi^{\circ}_{*}(f^{\circ})^{*}\mathcal{O}_{\mathbb{P}^{r}}(a)$
is a vector bundle of rank $ad$.

For a hypersurface $X\subseteq\mathbb{P}^{r}$ of degree $a$, the main
component is defined by the Cartesian diagram

${\overline{\mathcal{M}}_{1,n}(X,d)^{\text{main}}}$${\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)^{\text{main}}}$${\overline{\mathcal{M}}_{1,n}(X,d)}$${\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)}$

and similarly for the Vakil-Zinger desingularisation. Note that
$\mathcal{E}_{ad}$ gives a relative perfect obstruction theory for
$\widetilde{\mathcal{M}}_{1,n}(X,d)^{\text{main}}\hookrightarrow\widetilde{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)^{\text{main}}$,
hence a virtual class

(18)
$[\widetilde{\mathcal{M}}_{1,n}(X,d)^{\text{main}}]^{\mathrm{vir}}=c_{ad}(\mathcal{E}_{ad})\cap[\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)^{\text{main}}]$

of homological degree $d(r+1-a)+n$.

######  Definition 8.6.3.

For a hypersurface $X\subseteq\mathbb{P}^{r}$, the genus 1 _reduced
invariants_ are defined by integrating against
$[\widetilde{\mathcal{M}}_{1,n}(X,d)^{\text{main}}]^{\mathrm{vir}}$.

######  Remark 8.6.4.

Formula (18) can be thought of as a hyperplane property for reduced
invariants. Reduced invariants of a CY hypersurface are computable via torus
localisation (see [Zinger]).

####  8.6.2. The Zinger/ Li–Zinger formula

The following formula has been proved by Zinger/ Li and Zinger [Zin09a],
[LZ09a], [Zin08], [Zin09b] in the symplectic category and by Chang and Li
[CL15] in the algebraic one.

######  Theorem 8.6.5.

Let $X$ be a projective threefold of degree $a$. Then

$GW_{1}(X,d)=GW_{1}(X,d)^{\text{red}}+\frac{2+d(a-5)}{24}GW_{0}(X,d)$

Reasons to believe this formula:

  * •

Reduced invariants in genus 1 capture the contribution of the main component
to ordinary GW invariants.

  * •

From
$\mathcal{D}(X)^{1}\simeq\overline{\mathcal{M}}_{1,1}\times\overline{\mathcal{M}}_{0,n+1}(X,d)$,
we get a projection

$q:\mathcal{D}(X)^{1}\to\overline{\mathcal{M}}_{0,n}(X,d)$

which forgets the elliptic curve and the point to which it is attached. The
spaces $\overline{\mathcal{M}}_{0,n}(X,d)$ and
$\overline{\mathcal{M}}_{1,n}(X,d)$ have the same virtual dimension, so one
could hope for some form of virtual push-forward to hold.

  * •

The coefficient can be computed in an ideal situation: the generic fiber is
$\overline{\mathcal{M}}_{1,1}\times R$, where $R\simeq\mathbb{P}^{1}$ is the
fiber of the universal curve over $\overline{\mathcal{M}}_{0,n}(X,d)$. The
obstruction bundle on $\overline{\mathcal{M}}_{1,1}\times R$ is
$\mathbb{E}^{\vee}\boxtimes N_{R/X}$. Then the second Chern class gives the
coefficient of $GW_{0}$ in the formula.

  * •

The other components $\mathcal{M}(X)^{\lambda}$, with $\lambda$ as in 8.6.1,
$\lambda\neq 1$ should not contribute by (virtual) dimensional reasons.

Reasons why the formula is hard to prove:

  * •

As discussed in Section 3 the cone of $\overline{\mathcal{M}}_{1,n}(X,d)$ is
not the union of cones of the components of
$\overline{\mathcal{M}}_{1,n}(X,d)$. Consequently it is hard to split the
virtual class on the components of the moduli space
$\overline{\mathcal{M}}_{1,n}(X,d)$.

  * •

We do not have a perfect obstruction theory for the boundary components: Let
$q$ denote the node which joins the elliptic curve $E$ and the rational tail
$R$. The vector bundle
$\mathbb{E}^{\vee}\boxtimes\operatorname{ev}_{q}^{*}N_{R/X}$ does not extend
to a vector bundle on $\widetilde{\mathcal{D}}(X)^{1}$ (namely when $q$ is a
ramification point for the map).

  * •

Since the space $\widetilde{\mathcal{M}}_{1,n}(X,d)$ is rather mysterious, it
is hard to deal with the components of its normal cone above. In particular,
the normal cone may have components supported at the intersection of the main
component and boundary components.

  * •

Although $\mathcal{D}(X)^{1}$ fits into the following Cartesian diagram

${\mathcal{D}(X)^{1}}$${\mathcal{D}(\mathbb{P}^{r})^{1}}$${\overline{\mathcal{M}}_{0,n}(X,d)}$${\overline{\mathcal{M}}_{0,n}(\mathbb{P}^{r},d)}$

we cannot apply of the virtual pushforward theorem. This is because we do not
know if
$\mathfrak{C}_{\overline{\mathcal{M}}_{1,n}(X,d)/\mathfrak{M}_{1,n}}|_{\mathcal{D}(X)^{1}}$
maps to $\mathfrak{C}_{\overline{\mathcal{M}}_{0,n}(X,d)/\mathfrak{M}_{0,n}}$.

######  Example 8.6.6.

Let us look at the components of the virtual class of
$\widetilde{M}_{1,n}(\mathbb{P}^{r},d)$. In the following we show that we have
a splitting of the virtual class

$[\widetilde{M}_{1,n}(\mathbb{P}^{r},d)]^{\mathrm{vir}}=[\widetilde{M}_{1,n}(\mathbb{P}^{r},d)^{\mathrm{main}}]+[\widetilde{M}_{1,n}(\mathbb{P}^{r},d)^{\lambda}]^{\mathrm{vir}},$

where the virtual class on the boundary of the moduli space is given by the
restriction of the normal cone
$\mathfrak{C}_{\widetilde{M}_{1,n}(\mathbb{P}^{r},d)/\widetilde{\mathfrak{M}}_{1,n}}$
inside $h^{1}/h^{0}(R\pi f^{*}T_{\mathbb{P}^{r}})$ to the boundary components.
By the local equations (16), the cone
$\mathfrak{C}_{\widetilde{M}_{1,n}(\mathbb{P}^{r},d)/\widetilde{\mathfrak{M}}_{1,n}}$
does not have embedded components supported on the intersection of the main
component with the boundary components. Let us denote the component of
$\mathfrak{C}_{\widetilde{M}_{1,n}(\mathbb{P}^{r},d)/\widetilde{\mathfrak{M}}_{1,n}}$
supported on the main component by
$\mathfrak{C}^{\mathrm{main}}_{\widetilde{M}_{1,n}(\mathbb{P}^{r},d)/\widetilde{\mathfrak{M}}_{1,n}}$
and similarly the restriction of $h^{1}/h^{0}(R\pi f^{*}T_{\mathbb{P}^{r}})$
to the main component by $h^{1}/h^{0}(R\pi
f^{*}T_{\mathbb{P}^{r}})^{\mathrm{main}}$. We need to show that

$\mathfrak{C}^{\mathrm{main}}_{\widetilde{M}_{1,n}(\mathbb{P}^{r},d)/\widetilde{\mathfrak{M}}_{1,n}}\hookrightarrow
h^{1}/h^{0}(R\pi f^{*}T_{\mathbb{P}^{r}})^{\mathrm{main}}$

and

$\mathfrak{C}_{\widetilde{M}_{1,n}(\mathbb{P}^{r},d)^{\mathrm{main}}/\widetilde{\mathfrak{M}}_{1,n}}\hookrightarrow
T_{\widetilde{M}_{1,n}^{\mathrm{main}}(\mathbb{P}^{r},d)/\widetilde{\mathfrak{M}}_{1,n}}$

give the same class. Let us look at the local picture. We have local morphisms

$E_{\widetilde{M}_{1,n}(\mathbb{P}^{r},d)^{\mathrm{main}}/V}\to
E_{\widetilde{M}_{1,n}(\mathbb{P}^{r},d)/V}$

which are multiplication by $\xi$. By the local equations of
$\widetilde{M}_{1,n}(\mathbb{P}^{r},d)$, we have that

$N_{\widetilde{M}_{1,n}(\mathbb{P}^{r},d)^{\mathrm{main}}/V}\simeq
E_{\widetilde{M}_{1,n}(\mathbb{P}^{r},d)^{\mathrm{main}}/V}$
$C_{\widetilde{M}_{1,n}(\mathbb{P}^{r},d)/V}^{\mathrm{main}}\simeq
E_{\widetilde{M}_{1,n}(\mathbb{P}^{r},d)/V}^{\mathrm{main}}.$

This shows that the two virtual classes are equal to
$[\widetilde{M}_{1,n}(\mathbb{P}^{r},d)^{\mathrm{main}}]$, so in particular
they agree.

One could also use Theorem 7.2.5, although it is somewhat overcomplicated.
From the local description we see that we have a global morphism:

$T_{\widetilde{M}_{1,n}^{\mathrm{main}}(\mathbb{P}^{r},d)/\widetilde{\mathfrak{M}}_{1,n}}\to
T_{\widetilde{M}_{1,n}(\mathbb{P}^{r},d)/\widetilde{\mathfrak{M}}_{1,n}}$

which maps $\mathfrak{C}^{\mathrm{main}}$ to
$\mathfrak{C}_{\widetilde{M}_{1,n}(\mathbb{P}^{r},d)/\widetilde{\mathfrak{M}}_{1,n}}$.
The local discussion also shows that we are under the hypothesis of Theorem
7.2.5, which gives the result.

######  Example 8.6.7.

Let us look at local equations of $\widetilde{M}_{1,n}(X,d)$ inside $V$. The
easiest case is when $X\simeq\mathbb{P}^{3}\subseteq\mathbb{P}^{4}$ is a
hyperplane. Without loss of generality we may assume that $\mathbb{P}^{3}$ is
given by $\\{z_{1}=0\\}$ in $\mathbb{P}^{4}$. Then

(19)
${U_{\overline{\mathcal{M}}_{1,n}(X,d)}}$${U_{\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)}}$${V}$$\scriptstyle{\\{y_{1}=x_{1,1}=\ldots=x_{d,1}=0\\}}$$\scriptstyle{\\{\xi
y_{1}=\ldots=\xi y_{4}=0\\}}$

Note that there are $d+1$ equations on the left, and 5 on the right, but
$\overline{\mathcal{M}}_{1,n}(X,d)$ only has virtual codimension $d+5$ in $V$.
This is because the equation $\xi y_{1}$ on the right becomes superfluous
after adjoining all the equations on the left.

######  Remark 8.6.8.

Let us forget that it is easy to compute the contribution of the boundary of
$\widetilde{M}_{1,n}(\mathbb{P}^{3},d)$ and try to use the geometry of
$\widetilde{M}_{1,n}(\mathbb{P}^{4},d)$ to deduce it.

We first introduce more notation. Let $\pi:\mathcal{C}_{0}\to
M_{0,n}(\mathbb{P}^{3},d)$ be the universal curve and $E_{d+1}$ the vector
bundle $\pi_{*}f^{*}\mathcal{O}_{\mathbb{P}^{3}}(1)$ on
$M_{0,n}(\mathbb{P}^{3},d)$. Let $\mathfrak{C}_{X}$ be the relative normal
cone
$\mathfrak{C}_{\widetilde{M}_{1,n}(\mathbb{P}^{3},d)/\widetilde{\mathfrak{M}}_{1,n}}$.
Let $W$ in $V$ be given by equations

$\langle y_{2}\xi,y_{3}\xi,y_{4}\xi\rangle$

and let $j:M_{1,n}(\mathbb{P}^{3},d)\to W$. Consider now the deformation of
$h^{1}/h^{0}(\widetilde{\pi}_{*}\widetilde{f}^{*}T_{\mathbb{P}^{4}})$ to

(20)
$\pi_{*}f^{*}N_{\mathbb{P}^{3}/\mathbb{P}^{4}}\oplus\left[\frac{h^{1}\widetilde{\pi}_{*}\widetilde{f}^{*}T_{\mathbb{P}^{3}}}{h^{0}\widetilde{\pi}_{*}\widetilde{f}^{*}T_{\mathbb{P}^{4}}}\right]$

on the boundary of $\widetilde{M}_{1,n}(\mathbb{P}^{3},d)$. Looking at the
limit of $\mathfrak{C}_{X}|_{\mathcal{D}(X)^{\lambda}}$ in the vector bundle
stack (20), we see from equations 19 that it is locally isomorphic to

$C_{\widetilde{M}_{1,n}(\mathbb{P}^{3},d)\cap V/W}\times
C_{W/V}|_{\mathcal{D}(X)^{\lambda}\cap V}.$

This shows that the contribution of $\mathfrak{C}_{X}$ on the boundary
satisfies the push forward property

$q_{*}[\mathcal{D}(X)^{\lambda}]^{\mathrm{vir}}=N^{\lambda}[M_{0,n}(\mathbb{P}^{3},d)],$

for some $N^{\lambda}\in\mathbb{Q}$. The situation is similar for
hypersurfaces of higher degree, but there are two main complications: the map
$q$ is not proper anymore and the equations on the left are much less clear.
In particular, we cannot know whether they are independent of the set of
equations on the right and we cannot find an analogous local description of
the limiting cone. These are the main reasons why the above idea does not
generalise to higher degree. In [BCM] we substantially modify the above proof.

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