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

等变Chow-Witt群与椭圆曲线模栈

Equivariant Chow-Witt groups and moduli stacks of elliptic curves

本文引入等变Chow-Witt群定义商栈的Chow-Witt群，计算稳定与光滑椭圆曲线模栈的Chow-Witt环，并给出新生成元的几何解释及$μ_{2n}$分类栈的相应结果。

The integral Chow ring of the stack of 1-pointed hyperelliptic curves

In this paper we give a complete description of the integral Chow ring of the stack $\mathscr{H}_{g,1}$ of 1-pointed hyperelliptic curves, lifting relations and generators from the Chow ring of $\mathscr{H}_g$. We also give a geometric interpretation for the generators.

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.

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.

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 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.

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 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.

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.

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 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.

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.

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。

Infinitesimal Theory of Chow Groups via K-theory, Cyclic Homology and the Relative Chern Character

We examine the tangent groups at the identity, and more generally the formal completions at the identity, of the Chow groups of algebraic cycles on a nonsingular quasiprojective algebraic variety over a field of characteristic zero. We settle a question recently raised by Mark Green and Phillip Griffiths concerning the existence of Bloch-Gersten-Quillen-type resolutions of algebraic K-theory sheaves on infinitesimal thickenings of nonsingular varieties, and the relationships between these sequences and their "tangent sequences," expressed in terms of absolute Kähler differentials. More general…

商栈的几何动机的Chow权结构

The chow weight structure for geometric motives of quotient stacks

本文在任意系数下为商栈[X/G]的几何动机导出范畴构造了Chow权结构，并证明其核恢复Chow动机范畴。

The Orbifold Cohomology Ring of Simplicial Toric Stack Bundles

We introduce extended toric Deligne-Mumford stacks. We use an extended toric Deligne-Mumford stack to get the toric stack bundle and compute its orbifold Chow ring. Finally we generalize one result of Borisov, Chen and Smith so that the orbifold Chow ring of the toric stack bundle and the Chow ring of its crepant resolution are fibres of a flat family.

高维Chow循环、循环三次四维流形与拉格朗日子变种

Higher Chow cycles, cyclic cubic fourfolds and Lagrangian subvarieties

本文研究了holomorphic李代数流形上的高维Chow循环，通过构造显式不可分解的(2,1)和(4,1)循环，首次在holomorphic李代数流形上提供了此类循环的例子，并探讨了循环在拉格朗日子变种上的限制现象。

Chow rings of stacks of prestable curves II

We continue the study of the Chow ring of the moduli stack $\mathfrak{M}_{g,n}$ of prestable curves begun in [arXiv:2012.09887v2]. In genus $0$, we show that the Chow ring of $\mathfrak{M}_{0,n}$ coincides with the tautological ring and give a complete description in terms of (additive) generators and relations. This generalizes earlier results by Keel and Kontsevich-Manin for the spaces of stable curves. Our argument uses the boundary stratification of the moduli stack together with the study of the first higher Chow groups of the strata, in particular providing a new proof of the results of …

四次曲线模栈的Chow环和上同调环

Chow and cohomology rings of moduli stacks of plane quartics

本文研究了四次曲线模栈Hacking模栈的Chow环和上同调环，通过堆叠理论加权吹胀构造了一个光滑proper Deligne-Mumford堆栈，解决了KSBA和K-模空间之间的Calabi-Yau墙交叉问题，并计算了该模栈的Poincaré多项式，证明了循环类映射在有理系数下是同构的，并确定了其Chow环的生成元和关系。

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…

Towards an Intersection Chow Cohomology Theory for GIT Quotients

We study the Fulton-Macpherson operational Chow rings of good moduli spaces of properly stable, smooth, Artin stacks. Such spaces are étale locally isomorphic to geometric invariant theory quotients of affine schemes, and are therefore natural extensions of GIT quotients. Our main result is that, with rational coefficients, every operational class can be represented by a so-called topologically strong cycle on the corresponding stack. Moreover, this cycle is unique modulo rational equivalence on the stack. Using out methods, we prove that if $X$ is the good moduli space of a properly stable, s…

Chow rings of stacks of prestable curves I

We study the Chow ring of the moduli stack $\mathfrak{M}_{g,n}$ of prestable curves and define the notion of tautological classes on this stack. We extend formulas for intersection products and functoriality of tautological classes under natural morphisms from the case of the tautological ring of the moduli space $\bar{\mathcal{M}}_{g,n}$ of stable curves. This paper provides foundations for the second part of the paper. In the appendix (joint with J. Skowera), we develop the theory of a proper, but not necessary projective, pushforward of algebraic cycles. The proper pushforward is necessary …

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.

通过循环同调研究Chow群的变形

Deformations of Chow groups via cyclic homology

本文通过循环同调研究Chow群的变形，推广了Bloch等人的结果，解决了Chow群pro-可代表性与Hodge理论消失条件之间的联系问题。

点超椭圆曲线堆栈的积分Chow环

The Integral Chow Ring of the Stack of Pointed Hyperelliptic Curves

本文研究了参数化n个标记光滑超椭圆曲线的堆栈H_{g,n}的积分Chow环，对n=1,2完全计算，n≥3时至多计算二阶类的加法阶数，并在n=2g+3时获得部分结果，特别当g=2时，结果适用于M_{2,n}的Chow环。

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

In this paper we compute the Chow ring of the moduli stack $\bar{M}_2$ of stable curves of genus 2 with integral coefficients.

A Hochschild-cyclic approach to additive higher Chow cycles

Over a field of characteristic zero, we introduce two motivic operations on additive higher Chow cycles: analogues of the Connes boundary $B$ operator and the shuffle product on Hochschild complexes. The former allows us to apply the formalism of mixed complexes to additive Chow complexes building a bridge between additive higher Chow theory and additive $K$-theory. The latter induces a wedge product on additive Chow groups for which we show that the Connes operator is a graded derivation for the wedge product using a variation of a Totaro's cycle. Hence, the additive higher Chow groups with t…

平面结点曲线栈的周环

Chow ring of the stack of plane nodal curves

本文计算固定次数平面结点曲线模空间栈的有理周环并用重言类刻画，同时把 Vial 关于 Brauer–Severi 簇周群的结果推广至 $G$-等变情形。

带尖点超椭圆曲线栈上除子的周类

Chow classes of divisors on stacks of pointed hyperelliptic curves

本文在带尖点超椭圆曲线栈中计算通用 Weierstrass 除子与通用 $g^1_2$ 除子的周类，并用 Scavia 给出的除子类群基加以表达。

The intersection theory of the moduli stack of vector bundles on $\mathbb{P}^1$

We determine the integral Chow and cohomology rings of the moduli stack $\mathcal{B}_{r,d}$ of rank $r$, degree $d$ vector bundles on $\mathbb{P}^1$ bundles. We first show that the rational Chow ring $A_{\mathbb{Q}}^*(\mathcal{B}_{r,d})$ is a free $\mathbb{Q}$-algebra on $2r+1$ generators. The isomorphism class of this ring happens to be independent of $d$. Then, we prove that the integral Chow ring $A^*(\mathcal{B}_{r,d})$ is torsion-free and provide multiplicative generators for $A^*(\mathcal{B}_{r,d})$ as a subring of $A_{\mathbb{Q}}^*(\mathcal{B}_{r,d})$. From this description, we see that…

二阶向量丛在双曲曲线上的 universal moduli 堆叠中的循环

Cycles in the universal moduli stack of bundles of rank two over genus two curves

本文提出并证明了双曲曲线上的 universal moduli 堆叠的 Chow 环猜想，确定了生成元和关系，并计算了 genus 二曲线的 universal Jacobians 乘积的 Chow 环。

The Stack of Rational Nodal Curves

In this series of three papers we start to investigate the rational Chow ring of the stack consisting of nodal curves of genus 0, in particular we determine completely the rational Chow ring of the substack consisting of curves with at most 3 nodes. In this first paper we construct the stack of rational nodal curves and its stratification by nodes and show that the map from the universal curve to the stack is not representable in the category of schemes.

Rationality and Chow-Kuenneth decompositions for some moduli stacks of curves

In this paper, we show the existence of a Chow--Kuenneth decomposition for the moduli stack of stable curves of genus g with r marked points, for low values of g,r. We also look at the moduli space R of double covers of genus 3 curves, branched along 4 distinct points. We obtain a birational model of the moduli space R as a group quotient of a product of two Grassmanian varieties. This provides a Chow-Kuenneth decomposition over an open subset of R. The question of rationality of R is also discussed.

On the Chow ring of the classifying stack of PGL_3

We compute generators for the Chow ring of the classifying space of PGL_3 (over the complex numbers) as defined by Totaro. We also find enough relations after inverting 3. We show that this ring is not generated by Chern classes (this is the first example of this kind among classical groups) and prove that Totaro's refined cycle class map to a quotient of complex cobordism of BPGL_3 is surjective

超椭圆 Prym 对的模栈的整 Chow 环 I

The Integral Chow Rings of the Moduli Stacks of Hyperelliptic Prym Pairs I

本文是计算 Prym 对模栈整 Chow 环系列的第一篇，计算了由单对 Weierstrass 点生成，以及曲线亏格为奇数时至多 $(g-1)/2$ 对生成的 Prym 对的 Chow 环。

超椭圆Prym对的模栈的整Chow环 II

The Integral Chow Rings of the Moduli Stacks of Hyperelliptic Prym Pairs II

本文作为系列论文第二篇，计算了奇亏格下超椭圆Prym对模栈特定分支的整Chow环，还得到了奇亏格超椭圆Spin曲线模栈各不可约分支的相关结果，并给出了一个有独立价值的中间计算结果。

超椭圆 Prym 对模栈的积分 Chow 环 III

The Integral Chow Rings of the Moduli Stacks of Hyperelliptic Prym Pairs III

本文通过商栈描述和刚性化映射的根 gerbe 判别，完成了超椭圆 Prym 对模栈所有不可约分支积分 Chow 环的计算，仅剩一个奇亏格特殊情形未决。

Weierstrass 纤维化模空间的整系数 Chow 环

The integral Chow rings of moduli of Weierstrass fibrations

本文计算了射影直线上极小 Weierstrass 纤维化模空间的整系数 Chow 环，利用商栈表示和等变交叠理论确定了各 Chow 环的完整生成元与关系，并对 $N=1$ 和 $N=2$ 的情况给出了更明确的计算。

$\boldsymbol{\bar{\rm M}_{1,n}}$的穿墙整Chow环

Wall-crossing integral Chow rings of $\overline{\mathcal M}_{1,n}$

本文通过归纳计算Smyth提出的替代紧化的整Chow环，得到了$n=3,4$时$\bar{\text{M}}_{1,n}$的整Chow环，并指出$n\leq6$时该栈可由简单栈经加权胀缩得到。

On the Infinitesimal Theory of Chow Groups

The Chow groups of codimension-p algebraic cycles modulo rational equivalence on a smooth algebraic variety X have steadfastly resisted the efforts of algebraic geometers to fathom their structure. This book explores a "linearization" approach to this problem, focusing on the infinitesimal structure of the Chow groups near their identity elements. This method was adumbrated in recent work of Mark Green and Phillip Griffiths. Similar topics have been explored by Bloch, Stienstra, Hesselholt, Van der Kallen, and others. A famous formula of Bloch expresses the Chow groups as Zariski sheaf cohomol…

在$\mathbb{P}^1 \times \mathbb{P}^1$的循环覆盖上构造高 Chow 环，第二部分

Construction of higher Chow cycles on cyclic coverings of $\mathbb{P}^1 \times \mathbb{P}^1$, Part II

本文研究了在$\mathbb{P}^1 \times \mathbb{P}^1$的循环覆盖上构造高 Chow 环，并证明其生成不可分解部分的子群。

$\mathcal{M}_{1,n}$ 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 $10$。

循环覆盖上的高阶Chow环的构造$\mathbb{P}^1 \times \mathbb{P}^1$

Construction of Higher Chow cycles on cyclic coverings of $\mathbb{P}^1 \times \mathbb{P}^1$

本研究针对$\mathbb{P}^1 \times \mathbb{P}^1$的循环覆盖曲面族，构造了$(2, 1)$型高阶Chow环，通过计算超越调节映射的像，证明其在$\mathbb{Z}$上线性无关且生成$\operatorname{rank} \ge 36 \cdot φ(N)$的子群。

$A_r$-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 $\mathbb{Z}[1/6]$.

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。

超椭圆轨迹上的万有 Picard 层的 Chow 环

The Chow ring of the universal Picard stack over the hyperelliptic locus

本文确定了超椭圆轨迹上万有 Picard 层的有理 Chow 环及其生成元与关系，并计算了整 Picard 群，证明了其 Brauer 类阶由 $d-g$ 奇偶性决定。

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 (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.

Strong cycles and intersection products on good moduli spaces

We introduce conjectures relating the Chow ring of a smooth Artin stack $\mathcal{X}$ to the Chow groups of its possibly singular good moduli space $X$. In particular, we conjecture the existence of an intersection product on a subgroup of the full Chow group $A^*(X)$ coming from strong cycles on $\mathcal{X}$.

Arithmetic Intersection Theory on Deligne-Mumford Stacks

The arithmetic Chow groups and their product structure are extended from the category of regular arithmetic varieties to regular Deligne-Mumford stacks over the ring of integers in a number field.

Stacks of cyclic covers of projective spaces

We define stacks of uniform cyclic covers of Brauer-Severi schemes, proving that they can be realized as quotient stacks of open subsets of representations, and compute the Picard group for the open substacks parametrizing smooth uniform cyclic covers. Moreover, we give an analogous description for stacks parametrizing triple cyclic covers of Brauer-Severi schemes of rank 1, which are not necessarily uniform, and give a presentation of the Picard group for substacks corresponding to smooth triple cyclic covers.

Tautological Classes of the Stack of Rational Nodal Curves

This is the second in a series of three papers in which we investigate the rational Chow ring of the stack consisting of nodal curves of genus. Here we define the basic classes: the classes of strata and the Mumford classes.

On motives for Deligne-Mumford stacks

We define and compare two different definitions of Chow motives for Deligne-Mumford stacks, associated with two definitions of Chow rings. The main result we prove is that both categories of motives are equivalent to the usual category of motives of algebraic varieties, but the motives of a given stack associated with both theories are not isomorphic. We will also give some examples of motives associated with some alegbraic stacks.