The Integral Chow Ring of $\bar{M}_2$

Note on orbifold Chow ring of semi-projective toric Deligne-Mumford stacks

We prove a formula for the orbifold Chow ring of semi-projective toric DM stacks, generalizing the orbifold Chow ring formula of projective toric DM stacks by Borisov-Chen-Smith. We also consider a special kind of semi-projective toric DM stacks, the Lawrence toric DM stacks. We prove that the orbifold Chow ring of a Lawrence toric DM stack is isomorphic to the orbifold Chow ring of its associated hypertoric DM stack studied in \cite{JT}.

光滑非严格堆叠的积分Chow环

The Integral Chow Ring of Smooth Non-strict Toric Stacks

研究将Stanley-Reisner环扩展至非严格堆叠风扇，证明其与光滑非严格堆叠的积分Chow环同构，推广了更一般堆叠的Chow环结果。

Chow ring structure made simple

We show how to translate the task of computing the multiplicative structure of a Chow ring of a projective homogeneous variety into an easily understandable combinatorial task of calculating in the corresponding polynomial ring. The algorithms are also presented as a Maple package. Then we proceed to compute the multiplicative structure of the Chow rings for projective homogeneous varieties E6/P1, E7/P7, and E8/P8.

The Chow ring of a cubic hypersurface

We study the product structure on the Chow ring (with rational coefficients) of a cubic hypersurface in projective space and prove that the image of the product map is as small as possible.

A remark on the Chow ring of Küchle fourfolds of type $d3$

We prove that Küchle fourfolds $X$ of type d3 have a multiplicative Chow-Künneth decomposition. We present some consequences for the Chow ring of $X$.

$A_r$-stable curves and the Chow ring of $\overline{\mathcal{M}}_3$

In this work, we introduce the moduli stack $\widetilde{\mathcal{M}}_{g,n}^r$ of $n$-pointed, $A_r$-stable curves of genus $g$ and use it to compute the Chow ring of $\overline{\mathcal{M}}_3$. As a byproduct, we also compute the Chow ring of $\widetilde{\mathcal{M}}_3^7$. All the Chow rings are assumed to be with coefficients in $\mathbb{Z}[1/6]$.

On the Chow ring of a K3 surface

Though the Chow group of 0-cycles on a K3 surface is quite large, we observe that the subgroup generated by product of divisors is cyclic.

The integral Chow ring of the stack of smooth non-hyperelliptic curves of genus three

We compute the integral Chow ring of the stack of smooth, non-hyperelliptic curves of genus three. We obtain this result by computing the integral Chow ring of the stack of smooth plane quartics, by means of equivariant intersection theory.

The Chow ring of the symmetric space Gl(2n,C)/SO(2n,C)

We show this Chow ring is $\Z \oplus \Z$. We do this by partitioning the space into 2n subvarieties each of which is fibered over $Gl(2n-2,\C)/SO(2n-2,\C)$.

On the Chow ring of certain Fano fourfolds

We prove that certain Fano fourfolds of K3 type constructed by Fatighenti-Mongardi have a multiplicative Chow-Künneth decomposition. We present some consequences for the Chow ring of these fourfolds.

The Chow Ring of the Stack of Rational Curves with at most 3 Nodes

In this paper we explicit the rational Chow ring of the stack consisting of nodal curves of genus 0 with at most 3 nodes: it is a Q-algebra with 10 generators and 11 relations.

On the Chow ring of the stack of rational nodal curves

The goal of this paper is to compute the rational Chow ring of the stack consisting of nodal curves of genus 0 with at most 3 nodes: it is a Q-algebra with 10 generators and 11 relations.

On the Chow ring of birational irreducible symplectic varieties

We show that the graded Chow rings of two birational irreducible symplectic varieties are isomorphic. This lifts a result known for the cohomology algebras to the level of Chow rings, despite the non-injectivity the cycle class map. In the special case of general Mukai flops, we present an alternative approach based on explicit calculations.

The integral Chow ring of the stack of hyperelliptic curves of even genus

Let $g$ be an even positive integer. In this paper we compute the integral Chow ring of the stack of smooth hyperelliptic curves of genus $g$.

The Chow ring of the stack of hyperelliptic curves of odd genus

We find a new presentation of the stack of hyperelliptic curves of odd genus as a quotient stack and we use it to compute its integral Chow ring by means of equivariant intersection theory.

On the Chow ring of Fano varieties of type $S2$

We show that certain Fano eightfolds (obtained as hyperplane sections of an orthogonal Grassmannian, and studied by Ito-Miura-Okawa-Ueda and by Fatighenti-Mongardi) have a multiplicative Chow-Künneth decomposition. As a corollary, the Chow ring of these eightfolds behaves like that of K3 surfaces.

低亏格拟极化 K3 曲面模空间的周环

Chow ring of moduli spaces of quasi-polarised K3 surfaces in lower genus

本文研究低亏格拟极化 K3 曲面模空间，证明亏格不超过 5 时其周环由重言类构成。

The orbifold Chow ring of toric Deligne-Mumford stacks

Generalizing toric varieties, we introduce toric Deligne-Mumford stacks which correspond to combinatorial data. The main result in this paper is an explicit calculation of the orbifold Chow ring of a toric Deligne-Mumford stack. As an application, we prove that the orbifold Chow ring of the toric Deligne-Mumford stack associated to a simplicial toric variety is a flat deformation of (but is not necessarily isomorphic to) the Chow ring of a crepant resolution.

Inertial Chow rings of toric stacks

For any vector bundle $V$ on a toric Deligne-Mumford stack $\ix$ the formalism of \cite{EJK:16} defines two intertial products $\star_{V^{+}}$ and $\star_{V^{-}}$ on the Chow group of the inertia stack. We give an explicit presentation for the integral $\star_{V^+}$ and $\star_{V^-}$ Chow rings, extending earlier work of Boris-Chen-Smith \cite{BCS:05} and Jiang-Tsen \cite{JiTs:10} in the orbifold Chow ring case, which corresponds to $V = 0$. We also describe an {\em asymptotic} product on the rational Chow group of the inertia stack obtained by letting the rank of the bundle $V$ go to infinity…

The Chow Ring of the Stack of Smooth Plane Cubics

We give an explicit presentation of the integral Chow ring of a stack of smooth plane cubics. We also determine some relations in the general case of hypersurfaces of any dimension and degree.

On the Chow rings of classifying spaces for classical groups

We show how the stratification method, introduced by Vezzosi in his study of PGL_3, provides a unified approach to the known computations of the Chow rings of the classifying spaces of GL_n, SL_n, Sp_n, O_n and SO_n.

The Chow ring of a Fulton-MacPherson compactification

We give a short proof of a presentation of the Chow ring of the Fulton-MacPherson compactification of n points on an algebraic variety. The result can be found already in Fulton and MacPherson's original paper. However, there is an error in one of the lemmas used in their proof. In the process we also determine the Chow rings of weighted Fulton--MacPherson compactifications.

The Chow ring of Mbar_{0,m}(n, d)

We describe the Chow ring with rational coefficients of the moduli space of stable maps with marked points Mbar_{0,m}(n,d) as the subring of invariants of a ring B, relative to the action of the group of symmetries of d elements. B is computed by following a sequence of intermediate spaces for Mbar_{0,m}(n,d) and relating them to substrata of Mbar_{0,1}(n,d+m-1). An additive basis for the Chow ring is presented.

The Chow ring of relative Fulton-MacPherson space

Suppose that X is a nonsingular variety and D is a nonsingular proper subvariety. Configuration spaces of distinct and non-distinct n points in X away from D were constructed by the author and B. Kim in arXiv:0806.3819, by using the method of wonderful compactification. In this paper, we give an explicit presentation of Chow motives and Chow rings of these configuration spaces.

Quasi-symmetric functions and the Chow ring of the stack of expanded pairs

We show that the Hopf algebra of quasi-symmetric functions arises naturally as the integral Chow ring of the algebraic stack of expanded pairs originally described by J. Li, using a more combinatorial description in terms of configurations of line bundles. In particular, we exhibit a gluing map which gives rise to the comultiplication. We then apply the result to calculate the Chow rings of certain stacks of semistable curves.

The Chow ring of the stack of cyclic covers of the projective line

In this paper we compute the integral Chow ring of the stack of smooth uniform cyclic covers of the projective line and we give explicit generators.

Stable vector bundles as generators of the Chow ring

In this paper we show that the family of stable vector bundles gives a set of generators for the Chow ring, the K-theory and the derived category of any smooth projective variety.

$\mathcal{M}_{5,10}$ 的 Chow 环的有限性

Finiteness of the Chow ring of $\mathcal{M}_{5,10}$

本文证明了模空间 $\mathcal{M}_{5,10}$ 的有理 Chow 环作为 $\mathbb{Q}$-代数是有限生成的。

弦Chow环与加权爆破

Stringy Chow rings and weighted blow ups

本文在非代数闭基域上计算形如[X/G]的一般Deligne-Mumford栈的弦Chow环，并专门研究光滑簇沿光滑中心加权爆破的弦Chow环及其有限生成性质。

On the Chow ring of Fano varieties on the Fatighenti-Mongardi list

Conjecturally, Fano varieties of K3 type admit a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. We prove this for many of the families of Fano varieties of K3 type constructed by Fatighenti-Mongardi. This has interesting consequences for the Chow ring of these varieties.

$\overline{\mathcal{M}}_{1,3}$ 的积分 Chow 环

The integral Chow ring of $\overline{\mathcal{M}}_{1,3}$

本文利用多种拼接技术及带 $\ell$-adic 系数的高阶 Chow 群，计算了带三个标记点的稳定椭圆曲线模空间的积分 Chow 环。

On the Chow ring of some Lagrangian fibrations

Let $X$ be a hyperkähler variety admitting a Lagrangian fibration. Beauville's "splitting property" conjecture predicts that fibres of the Lagrangian fibration should have a particular behaviour in the Chow ring of $X$. We study this conjectural behaviour for two very classical examples of Lagrangian fibrations.

Cohomology ring of the flag variety vs Chow cohomology ring of the Gelfand-Zetlin toric variety

We compare the cohomology ring of the flag variety $FL_n$ and the Chow cohomology ring of the Gelfand-Zetlin toric variety $X_{GZ}$. We show that $H^*(FL_n, \mathbb{Q})$ is the Gorenstein quotient of the subalgebra $L$ of $A^*(X_{GZ}, \mathbb{Q})$ generated by degree $1$ elements. We compute these algebras for $n=3$ to see that, in general, the subalgebra $L$ does not have Poincare duality.

The Chow ring of double EPW sextics

A conjecture of Beauville and Voisin states that for an irreducible symplectic variety X, any polynomial relation between classes of divisors and the Chern classes of X which holds in cohomology already holds in the Chow groups. We verify the conjecture for a very general double EPW sextic.

An Additive Basis for the Chow Ring of \bar{M}_{0,2}(P^r,2)

We begin a study of the intersection theory of the moduli spaces of degree two stable maps from two-pointed rational curves to arbitrary-dimensional projective space. First we compute the Betti numbers of these spaces using Serre polynomial and equivariant Serre polynomial methods developed by E. Getzler and R. Pandharipande. Then, via the excision sequence, we compute an additive basis for their Chow rings in terms of Chow rings of nonlinear Grassmannians, which have been described by Pandharipande. The ring structure of one of these Chow rings is addressed in a sequel to this paper.

A remark on the Chow ring of some hyperkähler fourfolds

Let $X$ be a hyperkähler variety. Voisin has conjectured that the classes of Lagrangian constant cycle subvarieties in the Chow ring of $X$ should lie in a subring injecting into cohomology. We study this conjecture for the Fano variety of lines on a very general cubic fourfold.

Hilbert立方的Chow环

The Chow Ring of the Hilbert Cube

本文研究了Hilbert立方的Chow环，通过生成元和关系将其与X×Sym²(X)的Chow环联系起来，并在特征为零时扩展到准项目性情况。

$\mathcal{M}_{1,n}$ 当 $n=3,\dots,10$ 时的整 Chow 环

The integral Chow ring of $\mathcal{M}_{1,n}$ for $n=3,\dots,10$

本文计算了带有 $n$ 个标记点的光滑椭圆曲线模栈的整 Chow 环，其中 $n$ 的取值范围为 $3$ 到 $10$。

A remark on the Chow ring of Sicilian surfaces

We propose a "Bloch type" conjecture for surfaces: if the cup product map in coherent cohomology is zero, then all intersections of homologically trivial divisors should be zero in the Chow group of zero-cycles. We prove this conjecture for Sicilian surfaces.

On the Chow ring of Fano varieties of type $S6$

Fatighenti and Mongardi have defined Fano varieties of type S6 as zero loci of a certain vector bundle on the Grassmannian $\hbox{Gr}(2,10)$. These varieties have 3 Hodge structures of K3 type in their cohomology. We show that the Chow ring of these varieties also displays "K3 type" behaviour.

On the Chow ring of certain hypersurfaces in a Grassmannian

This small note is about Plücker hyperplane sections $X$ of the Grassmannian $\operatorname{Gr}(3,V_{10})$. Inspired by the analogy with cubic fourfolds, we prove that the only non-trivial Chow group of $X$ is generated by Grassmannians of type $\operatorname{Gr}(3,W_{6})$ contained in $X$. We also prove that a certain subring of the Chow ring of $X$ (containing all intersections of positive-codimensional subvarieties) injects into cohomology.

On the Chow Ring of the Classifying Space of some Chevalley Groups

Let $p$ be a prime and $q$ be a power of $p$. We compute the Chow ring of the classifying space of some Chevalley groups $G(\mathbb{F}_q)$, when considered as a finite algebraic group over a field of characteristic $p$ containing $\mathbb{F}_q$. Using specialization from characteristic $0$ to characteristic $p$ we also obtain results over the complex numbers.

The integral Chow ring of the stack of at most 1-nodal rational curves

We give a presentation for the stack of rational curves with at most 1 node as the quotient by GL(3) of an open set in a 6-dimensional irreducible representation. We then use equivariant intersection theory to calculate the integral Chow ring of this stack.

On the Chow ring of certain rational cohomology tori

Let $f: X \rightarrow A$ be an abelian cover from a complex algebraic variety with quotient singularities to an abelian variety. We show that $f^*$ induces an isomorphism between the rational cohomology rings $H^\bullet(A, \mathbb{Q})$ and $H^\bullet(X, \mathbb{Q})$ if and only if $f^*$ induces an isomorphism between the Chow rings with rational coefficients $\mathrm{CH}^\bullet(A)_{\mathbb{Q}}$ and $\mathrm{CH}^\bullet(X)_{\mathbb{Q}}$.

Grassmannian的Chow-Witt环

Chow-Witt rings of Grassmannians

本文计算了一般线性群分类空间的Chow-Witt环，涉及非平凡对偶性，并以此推导出有限Grassmannian的Chow-Witt环，所得公式与实Grassmannian整上同调环公式高度相似。

Equivariant operational Chow rings of T-linear schemes

We study $T$-linear schemes, a class of objects that includes spherical and Schubert varieties. We provide a localization theorem for the equivariant Chow cohomology of these schemes that does not depend on resolution of singularities. Furthermore, we give an explicit presentation of the equivariant Chow cohomology of possibly singular complete spherical varieties admitting a smooth equivariant envelope (e.g. group embeddings).

The (almost) integral Chow ring of $\overline{\mathcal{M}}_3$

This paper is the fourth in a series of four papers aiming to describe the (almost integral) Chow ring of $\overline{\mathcal{M}}_3$, the moduli stack of stable curves of genus $3$. In this paper, we finally compute the Chow ring of $\overline{\mathcal{M}}_3$ with $\mathbb{Z}[1/6]$-coefficients.

The rational Chow rings of moduli spaces of hyperelliptic curves with marked points

We determine the rational Chow ring of the moduli space $\mathcal{H}_{g,n}$ of $n$-pointed smooth hyperelliptic curves of genus $g$ when $n \leq 2g+6$. We also show that the Chow ring of the partial compactification $\mathcal{I}_{g,n}$, parametrizing $n$-pointed irreducible nodal hyperelliptic curves, is generated by tautological divisors. Along the way, we improve Casnati's result that $\mathcal{H}_{g,n}$ is rational for $n \leq 2g+8$ to show $\mathcal{H}_{g,n}$ is rational for $n \leq 3g+5$.

超椭圆 Weierstrass 点的叠的整 Chow 环

The integral Chow rings of the stacks of hyperelliptic Weierstrass points

本文计算了参数化带有 $n$ 个标记 Weierstrass 点的超椭圆曲线叠的整 Chow 环，并证明其作为代数可由任意 $Ψ$-类生成，所有关系均存在于 1 次中。

The (almost) integral Chow ring of $\widetilde{\mathcal{M}}_3^7$

This paper is the third in a series of four papers aiming to describe the (almost integral) Chow ring of $\overline{\mathcal{M}}_3$, the moduli stack of stable curves of genus $3$. In this paper, we compute the Chow ring of $\widetilde{\mathcal{M}}_3^7$ with $\mathbb{Z}[1/6]$-coefficients.

The Orbifold Chow Ring of Hypertoric Deligne-Mumford Stacks

Hypertoric varieties are determined by hyperplane arrangements. In this paper, we use stacky hyperplane arrangements to define the notion of hypertoric Deligne-Mumford stacks. Their orbifold Chow rings are computed. As an application, some examples related to crepant resolutions are discussed.

Chow ring of the moduli space of stable sheaves supported on quartic curves

Motivated by the computation of the BPS-invariants on a local Calabi-Yau threefold suggested by S. Katz, we compute the Chow ring and the cohomology ring of the moduli space of stable sheaves of Hilbert polynomial $4m+1$ on the projective plane. As a byproduct, we obtain the total Chern class and Euler characteristics of all line bundles, which provide a numerical data for the strange duality on the plane.

The Chow ring of the moduli space of curves of genus 6

We determine the Chow ring (with Q-coefficients) of M_6 by showing that all Chow classes are tautological. In particular, all algebraic cohomology is tautological, and the natural map from Chow to cohomology is injective. To demonstrate the utility of these methods, we also give quick derivations of the Chow groups of moduli spaces of curves of lower genus. The genus 6 case relies on the particularly beautiful Brill-Noether theory in this case, and in particular on a rank 5 vector bundle "relativizing" a baby case of a celebrated construction of Mukai, which we interpret as a subbundle of the …

On the Chow ring of the classifying stack of algebraic tori

We investigate the structure of the Chow ring of the classifying stacks $BT$ of algebraic tori, as it has been defined by B. Totaro. Some previous work of N. Karpenko, A. Merkurjev, S. Blinstein and F. Scavia has shed some light on the structure of such rings. In particular Karpenko showed the absence of torsion classes in the case of permutation tori, while Merkurjev and Blinstein described in a very effective way the second Chow group $A^2(BT)$ in the general case. Building on this work, Scavia exhibited an example where $A^2(BT)_\text{tors}\neq 0$. Here, by making use of a very elementary a…