Chow rings of stacks of prestable curves II
-
Younghan Bae
and
Johannes Schmitt
Abstract
We continue the study of the Chow ring of the moduli stack
1 Introduction
The tautological ring of the moduli stack of prestable curves
Let
of tautological classes, naturally extending the corresponding notion on
To describe the elements of
be the universal curve and let
be the i-th universal section and let
and for given
Let Γ be a prestable[1] graph in genus g with n markings. Each prestable graph defines a gluing map
(see e.g. [5, Section 2.1]). Given any prestable graph Γ, consider the products
of ψ and κ-classes on the space
Definition 1.1.
The tautological ring
The paper [5] then develops a calculus of decorated stratum classes. Below, such results from [5] are frequently referred.
In full generality, a description of the tautological ring
The tautological ring in genus zero
In Section 2, we give a complete description of the Chow groups of
For the moduli spaces
in
Later, Kontsevich and Manin [26, 27] showed that the Chow groups of
Generators
A first new phenomenon we see for
generated by the class
Instead, we prove that
Some decorated strata classes
In particular, since all such classes are contained in the tautological ring, we generalize Keel’s result that all Chow classes on
Theorem 1.2.
For
The idea of proof for this first theorem is easy to describe: consider the excision sequence of Chow groups for the open substack
From (1.5) for
we see that all classes in
One thing to verify in this last part of the argument is that the Chow group of a product of spaces
Proposition 1.3 (Proposition 2.6, Corollary 2.22).
Consider the stack
is surjective if Y has a good filtration by finite-type stacks and a stratification by quotient stacks. The map is an isomorphism if Y is a quotient stack.
In the proposition above, the technical conditions (like Y being a quotient stack or having a stratification by quotient stacks) are currently needed since some of the results we cite in our proof have them as assumptions. We expect that these conditions can be relaxed, but do not pursue this since Proposition 1.3 is sufficient for the purpose of our paper.
Relations
Returning to the stacks
on
Theorem 1.4 (informal version, see Theorems 2.31 and 2.33).
For
We give a precise description of what we mean by the system of relations “generated” by (1.4) and (1.8) in Definition 2.27, but roughly the allowed operations are as follows:
For
We can multiply relation (1.8) by an arbitrary polynomial in
Given a decorated stratum
See Example 2.28 for an illustration.
Again, our proof strategy for Theorem 1.4 begins by looking at the excision sequence (1.6), but now extended on the left using the first higher Chow group of
To illustrate how we can compute tautological relations using this sequence, consider the set of prestable graphs
For this purpose, we compute the first higher Chow groups of
the space
Restricting our argument to the moduli spaces
The proof relied on Keel’s result [22] that the WDVV relations generate the ideal of relations multiplicatively and thus required an explicit combinatorial analysis of the product structure of
In comparison, our proof is more conceptual, since we can trace each WDVV-relation on
A nontrivial consequence of our proof is the following result, stating that in codimension at least two the Chow groups of
Corollary 1.5 (see Corollary 2.37).
Let
of the boundary of
for
This result follows easily using higher Chow groups: we have the exact sequence
Using that
Relation to other work
Gromov–Witten theory
Gromov–Witten theory studies intersection numbers on the moduli spaces
results about the Chow groups of
As an example, in [17] Gathmann used the pullback formula of ψ-classes along the stabilization morphism
Chow rings of open substacks of
𝔐
0
,
n
Several people have studied Chow rings with rational coefficients of open substacks of
In [36], Oesinghaus computed the Chow rings of the loci
On the other hand, in [15, 13, 14] Fulghesu gave a computation of the Chow ring
Outlook and open questions
We want to finish the introduction with a discussion of some conjectures and questions about the Chow groups of
The first concerns the relation to the Chow groups of the moduli spaces
Conjecture (Conjecture 3.1).
Let
satisfies that the pullback
is injective.
It is easy to see that the system of morphisms
Since the map
Finally, recall that Theorems 1.2 and 1.4 completely determine the Chow rings of
For such open substacks U we can ask some more refined questions. The first concerns the structure of
Question 1 (Question 2.44).
Is it true that for
Supporting evidence for this question is that it has an affirmative answer for all stacks
Note that for U not of finite type, Question 1 will have a negative answer in general: from [36] it is easy to see that the Chow ring
Our second question concerns the Hilbert series
of the Chow ring of U.
Question 2 (Question 2.45).
Is it true that for
First note that a positive answer to Question 1 would imply Question 2 for all finite-type substacks
The Hilbert series of the Chow rings of open substacks U of
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Structure of the paper
In Section 2 we treat the Chow groups of the stacks
In Section 3 we compare the Chow rings of the stacks
Finally, Appendix A summarizes a construction of a Gysin pullback for higher Chow groups following [9, 24].
Notations and conventions
We work over an arbitrary base field k. For the convenience of the reader, we provide an overview of notations used in the paper in Table 2.
Notations
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moduli space of prestable curves |
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moduli space of prestable curves with values in a semigroup |
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moduli space of curves with dual graph precisely Γ |
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2 The Chow ring in genus 0
In this section we prove Theorem 1.2 and Theorem 1.4. These results completely describe the rational Chow group of
2.1 ψ and κ classes in genus 0
In [26, 27], Kontsevich and Manin described the Chow groups of
However the Chow group of the locus of smooth curves
Lemma 2.1.
For the moduli spaces of prestable curves in genus 0 we have
and
Proof.
The first three statements are proved in [13]. The last statement comes from the fact that
Note that for part (a) of the lemma above, it is important that we work with
so we see that there exists a nontrivial 2-torsion element in codimension 3.
By Lemma 2.1 we know that any monomial in κ- and ψ-classes on
We start with the ψ-classes. For
Lemma 2.2.
There is a codimension one relation
in
Proof.
Consider the
For the identification
We have that
it follows that
Consider the morphism
On
For
Now let
the class of the boundary divisor in
Lemma 2.3.
For
in
Proof.
When
and
via [5, Corollary 3.9]. ∎
For the κ-classes on
Lemma 2.4.
Let a be a nonnegative integer and consider
When
When
In the calculation below, we use the notion of tautological classes on the moduli stack
with additional decoration of
Proof.
It is enough to prove the corresponding statement on
(a) Consider the universal curve
so that
If
of
When
where implicitly we sum over all
Using the projection formula, the expression [5, Proposition 3.11] for the pushforward
(b) When a is an odd number, this statement follows from the Grothendieck–Riemann–Roch computation in [12, Proposition 1]. Namely,
where the sum is over
We prove the statement for
which is a composition of two forgetful maps
where
by Lemma 2.1. By the induction hypothesis, comparing the two equalities ends the proof. ∎
2.2 Generators of
CH
*
(
𝔐
0
,
n
)
The goal of this subsection is to prove Theorem 1.2.
The basic idea is simple: by Lemma 2.1 we know that classes on the smooth locus of
The two main technical steps to complete are as follows:
The boundary of
Knowing that classes on
The first issue is resolved by the fact that the pushforward along a proper surjective morphism of relative Deligne–Mumford type is surjective on the rational Chow group. We prove this statement in [5, Appendix B.4].
We now turn to the second issue, understanding the Chow group of products of spaces
Definition 2.5.
Let
is surjective, and we say that it has the Chow Künneth property (CKP) for Y if the map (2.2) is an isomorphism. Similarly, we define that X has the CKgP (or CKP) if it has the CKgP (or CKP) for all algebraic stacks Y of finite type over k.
Recall that a locally of finite-type stack X has a good filtration by finite-type stacks if X is the union of an increasing sequence
We now turn to showing the following result, resolving the second issue mentioned at the beginning of the section.
Proposition 2.6.
For all
For the proof we start with the smooth part of
Proposition 2.7.
For all
Proof.
Starting with the easy cases, for
for some
Next we consider the case
for
induced by pullback of the affine bundles are isomorphisms. The top arrow in the diagram is also an isomorphism since, as seen above,
We are left with the case
gives the universal curve over
induced by the
Now for any finite-type stack Y consider a commutative diagram
induced by the projective pushforward
Then the surjectivity of the top arrow follows.
To prove injectivity of the top arrow consider the diagram
induced by the flat pullback
Thus the map
For the next results, we say that an equidimensional, locally finite-type stack X has the Chow Künneth generation property up to codimension d if (2.2) is surjective in all codimensions up to d.
Lemma 2.8.
Let
Proof.
Fixing
and similarly
for any finite-type algebraic stack Y. This follows from the definition of the Chow groups as a limit (see the discussion above [5, Proposition A.5]). Using this we can reduce the proof of the lemma to the case where X (and similarly
Lemma 2.9.
Let Z be an algebraic stack, locally finite type over k, stratified by quotient stacks and with a good filtration by finite-type substacks. Let
If both Z and
Proof.
Let Y be an algebraic stack of finite type over k, stratified by quotient stacks. Then in the diagram
the top arrow is surjective by [5, Proposition B.19] (and [5, Remark B.21]) applied to the map
is surjective, so Z has the CKgP for stacks Y stratified by quotient stacks.
The statement with bounds on codimensions follows by looking at the correct graded parts of the above diagram and noting that codimension
Proposition 2.10.
Let X be an algebraic stack over k with a good filtration by finite-type substacks and let
If X is equidimensional and Z has pure codimension e, U has the CKgP up to codimension d and Z has the CKgP up to codimension
Proof.
For Y a finite-type stack, using excision exact sequences on X and
with exact rows. The vertical arrows for
Combining these ingredients, we are now ready to prove Proposition 2.6.
Proof of Proposition 2.6.
We will show that for all
by gluing maps. Note that
Proof of Theorem 1.2.
We show
By Lemma 2.1 all elements of
and by the induction hypothesis, all classes on the left side are (tensor products of) tautological classes up to degree
2.3 Higher Chow–Künneth property
The goal of this subsection is to give a background to compute the higher Chow group of
Below we study the Chow–Künneth property for higher Chow groups. Unlike the Chow–Künneth property for Chow groups, formulating the Chow–Künneth property for higher Chow groups in general is rather complicated, see [44, Theorem 7.2]. Below, we focus on the case of the first higher Chow group
Definition 2.11.
A quotient stack X over k is said to have the higher Chow Künneth property (hCKP) if for all algebraic stacks Y of finite type over k the natural morphism
is an isomorphism in total degree
Expanding this definition slightly, the degree
where
maps to
in the numerator of (2.6). The cokernel of the following map
is called the indecomposable part
We summarize some properties for higher Chow groups of quotient stacks
Lemma 2.12.
Let X be a quotient stack and
given by
is an isomorphism.
Proof.
Let
An affine bundle of rank r over X is a morphism
The complement of
We have a homotopy invariance property of higher Chow groups for affine bundles (see also [31, Proposition 2.3]).
Corollary 2.13.
Let X be a quotient stack and
is an isomorphism.
Proof.
Let p and q be projections from
because all stacks are quotient stacks [11].
Since
by [16, Lemma 3.3].
As α runs through a basis of
run through a basis of
form part of a basis of
gives an isomorphism.
The injectivity of
But since
Proposition 2.14.
For
is an isomorphism. In particular, setting
Proof.
When
When
Similarly,
for Y a quotient stack,
On the other hand, the natural morphism
is an isomorphism by Lemma 2.12. Combining with the equalities above, this shows that the map
is an isomorphism. This shows the hCKP of
When
for all quotient stacks Y. Then the hCKP and the vanishing
We are left with the case
induced by the projective pushforward
where E is a rank 3 vector bundle E on
be the relative hyperplane class. For any class α in
where the first equality comes from the functoriality of pushforward and Gysin pullback for higher Chow groups and the second equality is the projection formula (A.3) from Appendix A. The third equality comes from Lemma A.2 and the factor of two comes from the fact that the map
To prove injectivity of the top arrow consider the diagram
induced by the flat pullback
is injective and thus the left arrow of the above diagram is likewise injective. Hence the top arrow is an isomorphism, finishing the proof. ∎
The language of motives is a convenient way to state the higher Chow–Künneth property for
which sends a scheme to its motive.
The category
There is an invertible object, called the Tate motive
and by taking its shifting and tensor product we have
The motivic cohomology is a bi-graded module over the motivic cohomology of the base field k. The motivic cohomology of k is related to Milnor’s K-theory of fields.
In [45] Voevodsky proved that for any smooth scheme X over k, the higher Chow group and the motivic cohomology have the following comparison isomorphism:
where the right-hand side is Bloch’s higher Chow group introduced in [6]. Bloch’s definition of higher Chow groups will be used to compute the connecting homomorphism of the localization sequence. When X is a smooth scheme over k, the higher Chow group and the motivic cohomology have product structure
and the comparison isomorphism (2.12) is a ring isomorphism ([25]).
Now we summarize results from [8]. For a hyperplane complement
As a corollary,
The isomorphism
Example 2.15.
Let
and the element
Proposition 2.16.
Let
Proof.
Let Y be a quotient stack and hence
where the fifth equality comes from the cancellation theorem. In the proof of [8, Proposition 1.1], the index
After identifying
Proposition 2.14 and 2.16 compute the higher Chow group of
Now we revisit the CKP for the stack
be the algebraic m simplex. For
When
is given by specialization maps. If
where
Lemma 2.17.
Let
where
the boundary maps for the inclusions
For
where
The following diagram commutes:
(2.15)
where the arrow on the right is induced by the natural map
Proof.
(a) We first prove that the right-hand side is well-defined. For a different choice of extension
Then we have
This class vanishes because
We first prove the equality when
For each class in
and this proves the equality.
In general, let
is defined by the limit of boundary maps for naive higher Chow groups of
(b) Let
The same computation holds for
Remark 2.18.
By applying Lemma 2.17 to
vanishes since for
To prove the CKP for
Proposition 2.19.
Let X be an algebraic stack, locally of finite type over k with a good filtration and stratified by quotient stacks
Proof.
Since X has a good filtration, the Chow groups of X and
Now by assumption, there exists a nonempty open substack
where the columns are exact by the excision sequence. Since U has the CKP, the arrow
commutes by applying Lemma 2.17 to the map
To apply this to the stratification of
Lemma 2.20.
Let
from the Chow group of the quotient
is a surjection.
Proof.
The map π is representable and a principal G-bundle, hence in particular it is finite and étale. Thus we can both pull back cycles and push forward cycles under π.
For
shows that
thus restricted on the G-invariant part, we again have
showing
In the following, we typically apply the above lemma to the action of
Remark 2.21.
The above lemma is also true for the first higher Chow groups with
Corollary 2.22.
For all
Proof.
Recall that for a prestable graph Γ of genus 0 with n markings, there exists the locally closed substack
The product of spaces
We proved the Chow–Künneth property of
Remark 2.23.
Let X be an algebraic stack, locally of finite type over k. Let
which is compatible with projective pushforward, Chern classes and lci pullbacks. In [4], we will show that the cycle class map
2.4 Tautological relations
In this subsection, we formulate and prove a precise form of Theorem 1.4, see Theorem 2.31.
Recall that for a prestable graph Γ, a decoration α is an element of
Definition 2.24.
Define the strata space
By definition, the image of the map
is the tautological ring
For the proofs below, it is convenient to allow decorations
Definition 2.25.
Given a prestable graph Γ in genus 0 with n markings, an element
is said to be in normal form if
for vertices
for vertices
for vertices
for vertices
Note that because of the terms
Definition 2.26.
For
Definition 2.27.
Let
choosing a prestable graph Γ in genus g with n markings and a vertex
choosing an identification of the
choosing decorations
gluing the relation
More generally, given any family
On the level of Chow groups, the relations in Definition 2.27 are of the form
where
Example 2.28.
Let
For a prestable graph
and a decoration
Definition 2.29.
Consider the family
Define
Remark 2.30.
Let us comment on the role of the sets of relations appearing above. The relations
a (possibly zero) monomial term
a sum β of generators
The relations
Theorem 2.31.
The kernel of the surjection
In particular, we have
We split the proof of the above theorem into two parts.
Proposition 2.32.
The map
Proof.
The statement says that we can use
Theorem 2.33.
The kernel of the surjection
The proof is separated into several steps. The overall strategy is to stratify
and also
where
according to the number p of edges of graph Γ.[15] This descends to decompositions
for
From Proposition [5, Proposition 2.4] and Lemma 2.20 it follows that
Note that we have a natural map
Lemma 2.34.
The composition
is an isomorphism.
Proof.
First we note that
in which the vertical arrows are open embeddings, it follows that this is equivalent to first restricting α to
where the automorphisms σ act on
Now by Proposition 2.7 we have
Thus it follows from Lemma 2.1 that the set of all possible α such that
either the orbit contains
or the distinct elements in the orbit are linearly independent in
Basis elements
Next we realize the WDVV relation as the image of the connecting homomorphism
By [5, Proposition 2.4], the stack
Before we study the map
Lemma 2.35.
Let X be an equidimensional scheme and let
be two closed immersions. Consider the open embedding
Then we have a commutative diagram
where
Proof.
Elements of
On the other hand, to evaluate
Proposition 2.36.
For
is spanned by the set of WDVV relations, where we identify
Proof.
First, we prove this proposition when
Then
This is one of the WDVV relations on
The line
For the case of general
Note that the morphism π is flat and that we have an open embedding
and a closed embedding
Thus, since (under a suitable permutation of the markings) every generator of
is generated by WDVV relations on
We extend the above computation to
Corollary 2.37.
For
is the set of WDVV relations for
Proof.
By (2.13), we have
For the statement in degree
in other words by linear combinations of boundary divisors adding to zero in
There are exactly
in
To see this, we can simply construct test curves
Remark 2.38.
As a consequence of the above result, for
is an isomorphism in degree
is an isomorphism for
The surjectivity of
where
This corollary is enough to compute the connecting homomorphism in arbitrary degree.
Proposition 2.39.
The image of
in (2.23) is equal to the image of the composition
Thus we can write
Proof.
From the functoriality of higher Chow groups, it follows that we have a commutative diagram
where the sums run over prestable graphs with exactly p edges. By Remark 2.21,
From Remark 2.18 we know that
and thus factors through its cokernel. On the other hand, it follows from Propositions 2.14 and 2.16 that the cokernel of (2.26) is generated by classes coming from the direct sum
For an element
By Corollary 2.37, the elements
Proof of Theorem 2.33.
Recall that by Lemma 2.34, the composition
is an isomorphism. Thus in the diagram
we obtain a canonical splitting of the excision exact sequence (2.23) and thus we have
Combining (2.27) and equation (2.25) from Proposition 2.39, we see
Applying (2.28) for
finishing the proof. ∎
Finally, we end the proof of the main theorem.
Proof of Theorem 2.31.
We know that the kernel of
Likewise, by Theorem 2.33 we know that the kernel of
where the arrow
2.5 Relation to previous works
Let us start this subsection by pointing out several results in Gromov–Witten theory, studying intersection numbers on moduli spaces of stable maps, which can be seen as coming from results about the tautological ring of
Example 2.40.
In [32], degree one relations on the moduli space of stable maps to a projective space
Similarly, [32, Theorem 1.(1)] can be obtained from Lemma 2.3 on
From Theorem 2.31 we see that any universal relation in the Gromov–Witten theory of genus 0 obtained from tautological relations on
Apart from applications to Gromov–Witten theory, there are several results in the literature which compute the Chow groups of some strict open subloci of
Example 2.41.
Restricting to the locus
Example 2.42.
In [36], Oesinghaus computes the Chow ring (with integer coefficients) of the open substack
where we denote by
Under the isomorphism
Using the correspondence, we can verify several of the results of our paper in this particular example. Indeed, one can use Theorem 2.33 to verify that the classes (2.30) form a basis of
Note that [36] also computes the Chow group of the semistable loci
Example 2.43.
In a series of papers [15, 13, 14], Fulghesu presented a computation of the Chow ring of the open substack
Establishing a precise correspondence to the generators and relations discussed in our paper is challenging due to the complexity of the involved combinatorics. However, as a nontrivial check of our results we can compare the dimensions
which is the Hilbert series of the graded ring
In [14], Fulghesu has computed the Chow rings of the open substacks
The Hilbert series of the Chow rings of open substacks U of
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On the other hand, since for stable graphs with at most three edges no WDVV relations can appear, Theorem 2.33 implies that the Chow group
Case
recovering the formula from Table 3.
Case
Since the automorphism group of
where we used the substitution
Case
The automorphism group of
so the corresponding generator vanishes. Overall, the numbers of basis elements supported on
Adding this to the generating series for
again obtaining the same formula as in Table 3.
Case
giving a contribution of
to the generating series. The second type of generator is
Since
Due to the automorphism, we counted almost all the generators twice, except those fixed by the automorphism, for which
Adding these two series, we count every generator twice, so we obtain the correct count after dividing by two. Overall we get
However, expanding this series we obtain
Comparing with the expansion of the corresponding function
After revisiting Fulghesu’s proof, we think we can explain this discrepancy from a relation that was missed in [14]. In the notation of this paper, we claim that there is a relation
Here the classes
Our numerical experiments indicated that there are further relations missing in degrees
2.6 Chow rings of open substacks of
𝔐
0
,
n
– Finite generation and Hilbert series
In the previous subsection, we saw some explicit computations for Chow groups
For
(see [34, Theorem 13.2]).
This explains the shape of the Hilbert functions of
On the other hand, for the open substack
From this we can see two things:
The Chow ring
On the other hand, we still have that
The above observations lead to the following two questions.
Question 2.44.
Is it true that for
Question 2.45.
Is it true that for
For the first question, we note that by Theorem 1.2 we know that
For the second question, we observe that it would be implied for all finite-type open substacks U of
and
Since we know that the Hilbert series of
So Question 2.45 has a positive answer for the non-finite-type substacks of semistable points in
The rank of the Chow groups
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| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 2 | 3 | 4 | 6 | 11 | 23 | 50 | 108 |
| 2 | 3 | 5 | 9 | 16 | 33 | 80 | 215 | 621 | 1900 |
| 3 | 5 | 12 | 27 | 62 | 162 | 481 | 1572 | ||
| 4 | 13 | 32 | 84 | 235 | 739 | 2594 | |||
| 5 | 27 | 84 | 263 | 875 | 3219 | ||||
| 6 | 70 | 234 | 837 | 3219 | |||||
| 7 | 166 | 656 | 2683 | ||||||
| 8 | 438 | 1892 | |||||||
| 9 | 1135 | ||||||||
| 10 | 3081 |
To finish this section, we want to record some numerical data about the Chow groups of the full stacks
3 Comparison with the tautological ring of the moduli of stable curves
3.1 Injectivity of pullback by forgetful charts
Assume we are in the stable range
Restricting to the subrings of tautological classes, we note that the tautological ring
Thus the tautological ring
is the identity and thus
It is an interesting question whether the converse is true: do the Chow (or tautological) rings of the moduli spaces of stable curves determine the Chow (or tautological) ring of
Conjecture 3.1.
Let
satisfies that the pullback
is injective.
We have seen in [5, Lemma 2.1] that (for m sufficiently large) the image of
so certainly the image of
One aspect of the conjecture we can prove so far is the statement that if
Proposition 3.2.
For
Proof.
It suffices to show the statement for
Here π is the usual map forgetting the marking
Let
We want to transport this to an equality of classes on
Pushing forward by π and using
Thus, since
Again, for
The ranks of the Chow groups
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1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |||
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1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||
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3 | 1 | 2 | 3 | 3 | 3 | |||||
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5 | 1 | 2 | 4 | 5 | ||||||
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13 | 1 | 2 | 7 | |||||||
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1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
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2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | |||
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5 | 1 | 3 | 5 | 5 | 5 | 5 | ||||
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12 | 1 | 4 | 7 | 12 | 12 | |||||
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1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |||
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3 | 1 | 3 | 3 | 3 | 3 | 3 | ||||
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9 | 1 | 5 | 9 | 9 | 9 | |||||
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27 | 1 | 7 | 11 | 27 | ||||||
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1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |||
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4 | 1 | 4 | 4 | 4 | 4 | 4 | ||||
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16 | 1 | 5 | 15 | 16 | 16 | |||||
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62 | 1 | 5 | 16 | 62 | ||||||
3.2 The divisor group of
𝔐
g
,
n
As for the moduli space of stable curves, the group of divisor classes on
Thus we can restrict to the range
where
Proposition 3.3.
For
Proof.
As discussed before, for the statement that all divisor classes are tautological we can restrict to the stable range
is contained in
To compute the set of relations, we observe that
because
where the composition is an isomorphism. This shows that
It follows that all relations between decorated strata classes in codimension 1 are pulled back from
3.3 Zero cycles on
𝔐
g
,
n
After treating the case of codimension 1 cycles in the previous section, we want to make some remarks about cycles of dimension 0. For the moduli spaces of stable curves, these exhibit many interesting properties:
In [39], Pandharipande and the second author presented geometric conditions on stable curves
We want to note here, that for the moduli stacks of prestable curves, the behavior of tautological zero cycles becomes more complicated:
For
As visible from Table 4, the group
This indicates that for the moduli stacks of prestable curves, the group of zero cycles plays less of a special role than for the moduli spaces of stable curves.
Funding source: European Research Council
Award Identifier / Grant number: ERC-2017-AdG-786580-MACI
Award Identifier / Grant number: 786580
Funding statement: Younghan Bae was supported by ERC Grant ERC-2017-AdG-786580-MACI and Korea Foundation for Advanced Studies (KFAS). Johannes Schmitt was supported by the SNF Early Postdoc.Mobility grant 184245. The project has received funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (grant agreement No. 786580).
A Gysin pullback for higher Chow groups
In [9], Déglise, Jin and Khan generalized Gysin pullback along a regular imbedding to motivic homotopy theories. We summarize the construction in the language of higher Chow groups. For a moment, let X be a quasi-projective scheme over k and we consider higher Chow groups with
and let
be the morphism defined by the exterior product. Let
which fits into the cartesian diagram
By [6, 7] we have the localization sequence
Definition A.1.
For a regular imbedding
as a composition of following morphisms:
where
This definition extends the Gysin pullback for Chow groups in the sense that it coincides with the Gysin pullback defined in [16] when
Given a line bundle
We want to note two basic compatibilities of this operation: firstly, given a proper morphism
Secondly, intersecting higher Chow cycles with a Cartier divisor has the same formula as for ordinary Chow groups.
Lemma A.2.
Let
Proof.
Consider the following cartesian diagram:
where
By the transverse base change formula [9, Proposition 2.4.2],
Now we can conclude the result because
The above construction can be extended to global quotient stacks without any difficulty. Let X be an equidimensional quasi-projective scheme with a linearized G-action. Applying the Borel construction developed in [11] yields the definition of higher Chow groups for
Acknowledgements
We are grateful to Andrew Kresch, Jakob Oesinghaus and Rahul Pandharipande for many interesting conversations. We also thank Lorenzo Mantovani, Alberto Merici, Hyeonjun Park and Maria Yakerson for helpful explanations concerning higher Chow groups. The first author would like to thank Adeel Khan who helped to understand the construction of Gysin pullback for higher Chow groups. The second author would like to thank the Max Planck Institute for Mathematics in Bonn for its hospitality. Both authors want to warmly thank the referees for a very careful reading, and many suggestions to improve the clarity of the text.
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Articles in the same Issue
- Frontmatter
- Blown-up toric surfaces with non-polyhedral effective cone
- Local convexity estimates for mean curvature flow
- Chow rings of stacks of prestable curves II
- Kuznetsov’s Fano threefold conjecture via K3 categories and enhanced group actions
- On the collapsing of Calabi–Yau manifolds and Kähler–Ricci flows
- Relative entropy of hypersurfaces in hyperbolic space
- Finiteness and duality for the cohomology of prismatic crystals
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Articles in the same Issue
- Frontmatter
- Blown-up toric surfaces with non-polyhedral effective cone
- Local convexity estimates for mean curvature flow
- Chow rings of stacks of prestable curves II
- Kuznetsov’s Fano threefold conjecture via K3 categories and enhanced group actions
- On the collapsing of Calabi–Yau manifolds and Kähler–Ricci flows
- Relative entropy of hypersurfaces in hyperbolic space
- Finiteness and duality for the cohomology of prismatic crystals
- Quasi-plurisubharmonic envelopes 3: Solving Monge–Ampère equations on hermitian manifolds
- Ample line bundles and generation time
