The (almost) integral Chow ring of ℳ̅3
-
Michele Pernice
Abstract
This paper is the fourth in a series of four papers aiming to describe the (almost integral) Chow ring of
1 Introduction
The geometry of the moduli spaces of curves has always been the subject of intensive investigations, because of its manifold implications, for instance in the study of families of curves.
One of the main aspects of this investigation is the intersection theory of these spaces, which has both enumerative and geometrical implication.
In his groundbreaking paper [16], Mumford introduced the intersection theory with rational coefficients for the moduli spaces of stable curves.
He also computed the Chow ring (with rational coefficients) of
Edidin and Graham introduced in [9] the intersection theory of global quotient stacks with integer coefficients.
It is a more refined invariant, but as expected, the computations for the Chow ring with integral coefficients of the moduli stack of curves are much harder than the ones with rational coefficients.
To date, the only complete description for the integral Chow ring of the moduli stack of stable curves is the case of
The aim of this series of papers is to describe the Chow ring with
Let 𝜅 be a base field of characteristic different from
where
𝐻 is the fundamental class of the hyperelliptic locus.
1 of them is in codimension 2,
5 of them are in codimension 3,
8 of them are in codimension 4,
1 of them is in codimension 5.
At the end of this paper, we explain how to compare the result of Faber with ours and comment about the information we lose in the process of tensoring with rational coefficients.
Stable
A
r
-curves and the strategy of the proof
The strategy for the computation is the same as that used in [5] for the integral Chow ring of
to get the complete description of the Chow ring of 𝒳. To this end, we make use of a patching technique which is at the heart of the Borel–Atiyah–Seigel–Quillen localization theorem, and it has been used by many authors in the study of equivariant cohomology, equivariant Chow ring and equivariant K-theory. See the introduction of [6] for a more detailed discussion.
In general, without information regarding the kernel of the pushforward of the closed immersion
Unfortunately, there is no hope that this condition is verified if 𝒵 is a Deligne–Mumford separated stack, because in this hypothesis the integral Chow ring is torsion above the dimension of the stack.
This follows from [9, Theorem 3.2].
This is exactly the reason that motivated the authors of [5] to introduce the stack of cuspidal stable curves, which is not a Deligne–Mumford separated stack because it has some positive-dimensional affine stabilizers.
However, introducing cusps is not enough in the case of
Future prospects
As pointed out in the introduction of [5], the limitations of this strategy are not clear.
It seems that the more singularities we add, the more it is likely that the gluing condition is verified.
However, adding more singularities implies that we have to eventually compute the relations coming from such loci, which can be hard.
Moreover, we are left with a difficult problem, namely to find the right stratification for these newly introduced stacks.
We hope that this strategy will be useful to study the intersection theory of
Outline of the paper
This is the fourth (and last) paper in the series. It focuses on computing the Chow ring of the moduli stack of stable curves of genus 3.
Specifically, Section 2 is dedicated to recalling the theory of
In Section 3, we find a list of generators for the ideal of relation coming from the closed complement of
In Section 4, we explicitly compute the relations using the gluing lemma, restricting the fundamental classes of the locally closed strata to the stratification used in [19] to compute the Chow ring of
Finally, we summarize the results in the main theorem in Section 5, where we discuss the connection between our description and Faber’s one of the rational Chow ring of
2 Preliminaries and strategy
In this section, we recall the definition of the moduli stack
Moduli stack of
A
r
-stable curves
Fix a non-negative integer 𝑟.
Let 𝑔 be an integer with
Let 𝑘 be an algebraically closed field and
with
Notice that an
We fix a base field 𝜅 where all the primes smaller than
Let 𝐶 be a connected, reduced, one-dimensional, proper scheme over an algebraically closed field.
Let 𝑝 be a rational point which is a singularity of
if
if
if
We can define
We recall the following description of
Recall that we have an open embedding
The usual definition of the Hodge bundle extends to our setting.
See [20, Proposition 2.4] for the proof.
As a consequence, we obtain a locally free sheaf
Furthermore, we recall the existence of the (minimal) contraction morphism. This is [20, Theorem 2.5].
We have a morphism of algebraic stacks
where
The hyperelliptic locus
We recall the definition of hyperelliptic
Let 𝐶 be an
We define
Now we introduce another description of
Let 𝒵 be a twisted nodal curve of genus 0 over an algebraically closed field.
We denote by
We say that
the morphism
We say that
if 𝑝 is a non-stacky node and
if 𝑝 is a smooth point and
We say that
if
if
The conditions listed above may be interpreted as properties of the induced cyclic cover
Let us define now the stack classifying these data.
We denote by
We recall the following result, which gives us an alternative description of
The fibered category
Finally, we recall the following theorem which is a consequence of [18, Proposition 2.23 and Section 3].
The moduli stack
The Chow ring of
M
̃
3
7
In this subsection, we describe briefly the strategy used for the computation of the Chow ring of
First of all, we recall the gluing lemma.
Let
Therefore, we have the following commutative diagram of rings:
where 𝑞 is just the quotient morphism.
In the situation above, the induced map
is surjective and
From now on, we refer to the condition
We can apply Lemma 2.12 to the following stratification of
where
Notice that we need the surjectivity of the pullback of the closed immersion to apply Lemma 2.12. For such stratification, it follows from the following results:
̵[19, Corollaries 2.14 and 2.22] for the closed immersion
[19, Propositions 4.3–4.4] for the closed immersion
[19, Propositions 5.5–5.6] for the closed immersion
[19, Proposition 6.2] for the closed immersion
For a more precise description of the strata and for the computations needed, see [19].
We recall the main theorem of [19], which describes
We have the following isomorphism:
where 𝐼 is generated by the following relations:
We have the following geometric description of the generators:
𝐻 is the fundamental class of the hyperelliptic locus
For an explicit description of the relations, see [19, Remark 7.4].
2.1 Strategy of the computation for
M
̄
3
The final goal of this paper is to compute the Chow ring of
to compute the Chow ring of
We recall the notation used in [20, Section 4].
Let
and we define
We now introduce an alternative to
To define
This is standard and it can be done by taking the zero locus of the first Fitting ideal of
A local computation shows that the geometric points of
Let us define
where the
The first reason we choose to work with
The stack
See [20, Proposition 4.1] for the proof.
The second reason is that we have an explicit description of
The stack
When 𝑛 is odd, things are a bit more complicated.
In fact,
We have the following descriptions in the odd case.
We have a finite étale cover of degree 2,
where
The stack
A similar statement is true for the other connected components. For the proof of the following result, see [20, Proposition 4.18].
We have a morphism
where
Finally, we explain why we can reduce to studying the stacks
In the case of
Proof
It follows from the fact that every
Consider now the proper morphisms
and their restrictions to
which are still proper; let
In the setting above, we have that
Proof
This is a direct consequence of [4, Lemma 3.3]. ∎
The previous lemma implies that we need to focus on finding the generators of the relations coming from the strata
3 The relations from the
A
n
-strata
In this section, we are going to describe abstractly the generators of
Before we start with the abstract computations, we need to recall some notation and results from [20, Section 3].
Denote by
Corollary 3.5 of [20] gives us an isomorphism between
Proposition 3.7 of [20] gives us a description of
where
Finally, we recall that the morphism
where
Generators for the image of
ρ
n
,
∗
if 𝑛 is even
We are going to prove that the morphism
is surjective for
The morphism
Proof
We start by considering the isomorphism proved in [20, Corollary 4.12] (see Proposition 2.16) in case
see Notation 3.1 for a description of
is cartesian.
The morphism
We want to understand the pullback of the hyperelliptic locus, i.e.
The morphism
Proof
Again, Proposition 2.16 shows that
where 𝑠 is the generator of the Picard group of
therefore, it is enough to prove that
We know that
The locus
Before going to study the morphism
Recall that
The Chow ring of
the 𝜆-classes
the 𝜓-class
two classes
a homogeneous polynomial of degree 7.
Proof
We do not describe all the computation in detail.
The idea is to use the stratification introduced in [5, Section 4], i.e.
Clearly,
We are ready for the proposition.
The image of the pushforward of
is generated by the elements
Proof
For this proof, we denote by
We need to describe the pullback of the generators of the Chow ring of
Notice that
Finally, to compute the restriction of
which implies that
Notice that
In the same way,
Generators for the image of
ρ
n
,
∗
if 𝑛 is odd
Now we deal with the odd case. This is a bit more convoluted as we have several strata to deal with for every 𝑛. Let us recall the descriptions we have. See Proposition 2.17 and Proposition 2.18.
First,
if
if
if
We start with the case
The pullback of the morphism
Proof
The proof is similar to the one of Proposition 3.2.
First of all, we describe the Chow ring of
where the invariants are taken with respect to the action of
By construction, we have that
where the morphism 𝑓 is described in the proof of Proposition 3.2 (see also Notation 3.1).
A simple computation shows that
We now deal with the case
The pullback of the morphism
Proof
In this case, Proposition 2.18 tells us that
where the morphism
where 𝑢 (respectively 𝑠) is the generator of the Chow ring of
Clearly,
Moreover, as the closed embedding
and the induced long exact sequence on the global sections
We know that
where as usual
Finally, a proof similar to the one of Proposition 3.8 gives us the following result.
The pullback of the morphism
Proof
We leave it to the reader to check the details. See also Proposition 3.12 for a similar result in the non-separating case. ∎
It remains to prove the case for
The pullback of the morphism
Proof
Thanks to Proposition 2.18, we can describe
is an isomorphism at the level of Chow rings, it is enough to understand the
We recall that
The
Furthermore, the
where
Finally, we arrived at the end of this sequence of abstract computations.
The pullback of the morphism
Proof
For this proof, we denote by
First of all, we know that
see [5, Proposition 3.3].
It is important to remark that
We need to understand the class of the
This implies that the class associated to the torsor
where
Lemma 2.20 implies that the image of
the fundamental classes of
the fundamental classes of the images of
the fundamental classes of the closure of the images of
Notice that
The Chow ring of
4 Explicit description of the relations
We illustrate the strategy to compute the explicit description of the relations listed in Corollary 3.14.
Suppose we want to compute the fundamental class of a closed substack 𝑋 of
Once we have the explicit descriptions, we need to patch them together.
Let us show how to do it for the first two strata, i.e.
an expression of 𝑋 (which makes sense because of Remark 2.13), we need to compute the two polynomials 𝑝 and 𝑞.
If we restrict 𝑋 to
where as usual
Before proceeding with the computation, we recall the description of the Chow rings of the strata.
The Chow ring of
The Chow ring of
There is a
which gives us that
where
There is an
which gives us that
where
For the rest of the section, we will follow the notation set in Remark 4.1 for the generators of the Chow rings of the strata.
We show how to apply this strategy firstly for the computations of the fundamental classes
We have the following description:
and
in the Chow ring of
Proof
First of all, we have that
which implies that
Now we have to compute the restriction of
is not regular.
As a matter of fact, one can prove that doing the naive computation does not work, i.e. the difference
where
is described in the proof of Lemma 3.4 and
the normal bundle
where 𝑡 is the generator of
is commutative, where
A simple computation using the fact that
Using the explicit description of the pullback
Finally, we have to restrict
where the right square is cartesian, the morphism
is the closed immersion described in [5, Section 4] (see also the proof of Lemma 3.4), the morphism
moreover, we have that
where
in the Chow ring of
∎
The relation
The fundamental class of
A
5
0
and
A
3
1
Now we concentrate on describing two of the strata of separating singularities, namely
We have the description
in the Chow ring of
Proof
A generic object in
Consider now the description of
we have that excess intersection theory gives us the equalities
where
Let us focus on the restriction to
described in Theorem 2.6.
Through this identification,
The same idea works for
Finally, it is clear that
Now we focus on the fundamental class of
The intersection of
We have the equality
in the Chow ring of
Proof
Because every
see [19, Corollary 4.2] for a more detailed discussion.
We have that an element
It remains to describe the restriction of
where 𝑈 is an invariant open inside a
To describe
Recall that we have the isomorphism
Finally, let us denote by 𝐿 the standard representation of
In the setting above, we have the isomorphism
of algebraic stacks.
Proof
Thanks to the isomorphism
We denote it simply by
Consider an element 𝑝 in
where
We define 𝑊 as the 𝐻-representation
and we consider the 𝐻-equivariant closed embedding of
which is defined by the association
induced by the closed immersion
where the 𝐺-representation
of stacks. ∎
By construction, an element
We have an isomorphism of rings
Let 𝑋 be the closed substack of
where
The closed immersion
defined by equations
whose fundamental class is equal to
Moreover, the morphism
Proof
This follows from the proof of Lemma 4.6.∎
Because
We are finally ready to do the computation.
The closed substack
Proof
The element of the representation
A straightforward computation gives us the fundamental class of
Fundamental class of
A
n
-singularity
We finally deal with the computation of the remaining fundamental classes. As usual, our strategy assures us that it is enough to compute the restriction of every fundamental class to every stratum. We do not give details for every fundamental class. We instead describe the strategy to compute all of them in every stratum and leave the remaining computations to the reader.
We start with the open stratum
First of all, we can reduce the computation to the fundamental class of the locus
In the situation above, we have that
where
while
for every
Proof
Proposition 4.9 and Lemma 4.6 imply that it is enough to compute the fundamental class of
Therefore, we need to find a relation between the coefficients of
To do so, we apply Weierstrass preparation theorem.
Specifically, we use [10, Algorithm 5.2], which allows us to write the polynomial
Although
Notice that, for
It remains to compute the fundamental class of
The restriction of
for
As far as
The restriction of
for
We now concentrate on the stratum
as in [19, Section 4] and we denote by
We have the following description of the
the fundamental classes of
the fundamental class of
the fundamental class of
the fundamental class of
Proof
If
The last part of the computations is the restriction to the hyperelliptic locus.
We recall the stratification
Lemma 2.12 implies that we can compute the restriction of
The restriction of
Proof
This is an easy exercise and we leave it to the reader. ∎
Restriction to
H
̃
3
∖
(
Ξ
2
∪
Δ
̃
1
)
It remains to compute the restriction of
We know that
Therefore, we have that
Using the description of
5 The Chow ring of
M
̄
3
and the comparison with Faber’s result
We are finally ready to present our description of the Chow ring of
Let 𝜅 be the base field of characteristic different from
where
We write the relations explicitly as follows:
Notice that the relations
Lastly, we compare our result with the one of Faber, namely [12, Theorem 5.2].
Recall that he described the Chow ring of
First of all, if we invert 7, we have that the relation
which explains how to pass from the generator
Thus we can construct two morphisms of ℚ-algebras,
which are one the inverse of the other. A computation shows that 𝜙 sends our ideal of relations to the one in [12] and 𝜑 sends the ideal of relations in [12] to the one we constructed.
A Discriminant relations
In this appendix, we generalize [8, Proposition 4.2]. We do not need this result in its full generality in our work, only the formulas in Remark A.7.
First of all, we set some notation.
Everything is considered over a base field 𝜅.
Let 𝑇 be the 2-dimensional split torus
Let 𝑁 and 𝑘 be two positive integers such that
We want to study the image of the pushforward of the closed immersion
where
For
Exactly as it was done in [22] and generalized in [7], we introduce the multiplication morphism for every positive integer 𝑟 such that
Suppose that the characteristic of 𝜅 is greater than 𝑁.
Then the disjoint union of the morphisms
Therefore, it is enough to study the image of the pushforward of
Fix
in
with
Therefore, it is enough to describe the ideal generated by
We define the element associated to
Our goal is to prove that the ideal is in fact generated by
Let 𝑛 be an integer and
the small diagonal in the 𝑛-fold product, i.e. the morphism defined by the association
We denote by
Notice that we are using the same notation for two different elements: if we are in the projective space
We have a commutative diagram of finite morphisms
where
We have the identity
in the Chow ring of
Proof
The diagonal
Notice that, in the Chow ring of
The case
and thus, by induction,
Recall that we have the relations
between elementary symmetric functions (with
where we used the relation
Shifting the index of the last sum, it is easy to get the following identity:
and the statement follows from shifting the last sum again and from using the relations between the symmetric functions (notice that
Define by
induced by the 𝑇-equivariant closed immersion
for every
From now on, we set
We denote by
and by
We get the equality
in the 𝑇-equivariant Chow ring of
Proof
Thanks to Lemma A.4, we have that
we need to take the image through
for every 𝑠-tuple
The statement follows. ∎
Notice that the expression makes sense also per
Let us describe the case
if
These two formula give us the 𝑇-equivariant class of these two elements.
As matter of fact,
We can describe
We denote by 𝐼 the ideal generated by the two elements described in the previous remark. First of all, we prove that almost all the pushforwards we need to compute are in this ideal.
We have that
for every
Proof
Lemma A.6 implies that it is enough to prove that
Therefore, it only remains to prove that
We have the equality
in the 𝑇-equivariant Chow ring of
Proof
Denote by
which gives us that
Before going forward with our computation, we recall the following combinatorial fact.
For every pair of non-negative integers
We are going to use it to prove the following result.
For every non-negative integer
in the 𝑇-equivariant Chow ring of
Proof
The left-hand side of the equation in the statement can be written as
see Remark A.7. If we apply Proposition A.9 to the two sums, we get
if we exchange the sums in each factor and put everything together, we end up with
Shifting the inner sum and setting
Notice that we can extend the inner sum up to
and we can conclude using Lemma A.10. ∎
We state now the last technical lemma.
If we define
in the 𝑇-equivariant Chow ring of
Proof
We proceed by induction on
Suppose that
we can apply the previous proposition again and get
which is the same as
The statement follows by induction. ∎
It is important to notice that more is true; the same exact proof shows us that
for every
Before going to prove the final proposition, we recall the following combinatorial fact.
We have the numerical equality
for every
for
Finally, we are ready to prove the last statement.
We have
Proof
Notice that
therefore, we have to study the first
therefore, it remains to prove that the coefficient
is an integer for every
which implies that, for
is an integer.
Notice that it is clearly true for
The image of
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© 2024 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
Articles in the same Issue
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- Jacobian determinants for nonlinear gradient of planar ∞-harmonic functions and applications
- Fano varieties with torsion in the third cohomology group
- The distribution of Manin’s iterated integrals of modular forms
- On some 𝑝-adic and mod 𝑝 representations of quaternion algebra over ℚ𝑝
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- Simple 𝑝-adic Lie groups with abelian Lie algebras
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Articles in the same Issue
- Frontmatter
- The theory of F-rational signature
- Jacobian determinants for nonlinear gradient of planar ∞-harmonic functions and applications
- Fano varieties with torsion in the third cohomology group
- The distribution of Manin’s iterated integrals of modular forms
- On some 𝑝-adic and mod 𝑝 representations of quaternion algebra over ℚ𝑝
- Nonexistence of isoperimetric sets in spaces of positive curvature
- Simple 𝑝-adic Lie groups with abelian Lie algebras
- Hyperbolic lattice point counting in unbounded rank
- The (almost) integral Chow ring of ℳ̅3
