Alexander Polishchuk
Let $J$ be the Jacobian of a smooth projective curve. We define a natural action of the Lie algebra of polynomial Hamiltonian vector fields on the plane, vanishing at the origin, on the Chow group of $J$ with rational coefficients. Using this action we obtain some relations between tautological cycles on $J$.
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We study subrings in the Chow ring $\CH^*(A)_{\Bbb Q}$ of an abelian variety $A$, stable under the Fourier transform with respect to an arbitrary polarization. We prove that by taking Pontryagin products of classes of dimension $\leq 1$ one gets such a subring. We also show how to construct finite-dimensional Fourier-stable subrings in $\CH^*(A)_{\Bbb Q}$. Another result concerns the relation between the Pontryagin product and the usual product on the $\CH^*(A)_{\Bbb Q}$. We prove that the operator of the usual product with a cycle is a differential operator with respect to the Pontryagin prod…
On the Chow and cohomology rings of moduli spaces of stable curves
本文研究稳定曲线模空间 $\overline{\mathcal{M}}_{g,n}$ 的有理 Chow 环与上同调环何时由赘言类生成，并证明若干低亏格、低标记点情形下二者同构且为赘言环。
We define tautological Chow classes on the moduli space of curves with linear series. In the case where the forgetful morphism to the moduli space of curves has relative dimension zero, we describe the images of these classes in the Chow group of Mgbar. As an application, we compute the (virtual) slopes of several different classes of divisors on Mgbar.
We determine a subring of the Chow ring and the cohomology of $BPGL_n$, the classifying space of the projective linear group of degree $n$ over complex numbers, and explain a way in which this computation might play a role in the period-index problem. In addition, we show that the Chow ring of $BPGL_n$ is not generated by the Chern classes of linear representations of $PGL_n$.
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 …
Let $\mathcal{H}_{k,g}$ be the Hurwitz stack parametrizing degree $k$, genus $g$ covers of $\mathbb{P}^1$. We define the tautological ring of $\mathcal{H}_{k,g}$ and we show that all Chow classes, except possibly those supported on the locus of "factoring covers," are tautological up to codimension roughly $g/k$ when $k \leq 5$. The set-up developed here is also used in our subsequent work, wherein we prove new results about the structure of the Chow ring for $k \leq 5$.
We describe a theory of logarithmic Chow rings and tautological subrings for logarithmically smooth algebraic stacks, via a generalisation of the notion of piecewise-polynomial functions. Using this machinery we prove that the double-double ramification cycle lies in the tautological subring of the (classical) Chow ring of the moduli space of curves, and that the logarithmic double ramification cycle is divisorial (as conjectured by Molcho, Pandharipande, and Schmitt).
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 …
The Chow ring of $\mathcal{S}_5^-$ is tautological
本文证明了亏格5的奇自旋曲线模空间$\mathcal{S}_5^-$的Chow环是重言的，通过分析典范亏格5曲线和全切超平面的几何性质，并证明了$\mathcal{M}_{5,4}$中支配$\mathcal{S}_5^-$的微分层的理性。
We study tautological cycle classes on the Jacobian of a curve. We prove a new result about the ring of tautological classes on a general curve that allows, among other things, easy dimension calculations and leads to some general results about the structure of this ring. Next we obtain a vanishing result for some of the generating classes p_i; this gives an improvement of an earlier result of Herbaut. Finally we lift a result of Herbaut and van der Geer-Kouvidakis to the Chow ring (as opposed to its quotient modulo algebraic equivalence) and we give a method to obtain further explicit cycle r…
In this paper we prove that the rational Chow rings of $\mathcal{M}_{5,8}$ and $\mathcal{M}_{5,9}$ are tautological, and that these moduli spaces have the Chow Künneth generation Property.
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.
The Chow ring of $\mathcal{M}_g$ is known to be generated by tautological classes for $g \leq 9$. Meanwhile, the first example of a non-tautological class on $\mathcal{M}_{g}$ is the fundamental class of the bielliptic locus in $\mathcal{M}_{12}$, due to van Zelm. It remains open if the Chow rings of $\mathcal{M}_{10}$ and $\mathcal{M}_{11}$ are generated by tautological classes. In these cases, a natural first place to look is at the bielliptic locus. In genus $10$, it is already known that classes supported on the bielliptic locus are tautological. Here, we prove that all classes supported o…
We present a collection of algebraic equivalences between tautological cycles on the Jacobian $J$ of a curve, i.e., cycles in the subring of the Chow ring of $J$ generated by the classes of certain standard subvarieties of $J$. These equivalences are universal in the sense that they hold for all curves of given genus. We show also that they are compatible with the action of the Fourier transform on tautological cycles and compute this action explicitly.
Let $M_g$ be the moduli space of smooth genus $g$ curves. We define a notion of Chow groups of $M_g$ with coefficients in a representation of $Sp(2g)$, and we define a subgroup of tautological classes in these Chow groups with twisted coefficients. Studying the tautological groups of $M_g$ with twisted coefficients is equivalent to studying the tautological rings of all fibered powers $C_g^n$ of the universal curve $C_g \to M_g$ simultaneously. By taking the direct sum over all irreducible representations of the symplectic group in fixed genus, one obtains the structure of a twisted commutativ…
Logarithmic tautological rings of the moduli spaces of curves
本文定义曲线模空间的对数重言环，借助同调分段多项式与锥叠组合完成亏格0计算，并将标准重言环关系及 Pixton 猜想提升至对数情形。
Chow rings, cohomology rings, and point counts of moduli spaces of curves
综述曲线模空间Chow环、上同调环和有限域点计数三个密切相关不变量的最新结果，探讨它们何时由tautological类生成。
Pushforward of Siegel flag varieties in the Chow ring
本文计算了 Siegel 情形下子旗簇在 Chow 环中的类，并指出其对应于阿贝尔簇紧化模空间 tautological 环中支撑在环面边界上的类生成元，猜想该对应关系并非巧合。
We exhibit a 2-dimensional family of non-hyperelliptic curves of genus 5, called Humbert curves, for which the tautological ring injects into cohomology. In particular, Humbert curves have a multiplicative Chow-Künneth decomposition (in the sense of Shen-Vial), and their Ceresa cycle is torsion.
We consider the Chow ring with rational coefficients of the Jacobian of a curve. Assume D is a divisor in a base point free g^r_d of the curve such that the canonical divisor K is a multiple of the divisor D. We find relations between tautological cycles. We give applications for curves having a degree d covering of P^1 whose ramification points are all of order d, and then for hyperelliptic curves.
Extension Groups of Tautological Bundles on Punctual Quot Schemes of Curves
本文证明了曲线上点态 Quot 概形上重言丛的上同调与扩张群公式，并由此推出重言丛决定原始向量丛的同构类，同时给出沿 Quot--Chow 态射推前的消失结果。
An introduction to the intersection theory of the moduli space of curves
介绍曲线模空间的相交理论及其环状结构，综述开放问题并概述证明周环由环状类生成与否的技术。
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.
On product identities and the Chow rings of holomorphic symplectic varieties
本文针对K3曲面上稳定层模空间提出推广Beauville-Voisin恒等式的猜想，并证明当模空间为Hilbert概形时猜想成立，同时探讨其对tautological子环结构的影响。
The virtual geometry of the moduli space of stable quotients is used to obtain Chow relations among the kappa classes on the moduli space of nonsingular genus g curves. In a series of steps, the stable quotient relations are rewritten in successively simpler forms. The final result is the proof of the Faber-Zagier relations (conjectured in 2000).
Extensions of tautological rings and motivic structures in the cohomology of $\overline{\mathcal{M}}_{g,n}$
本文研究 $\overline{\mathcal{M}}_{g,n}$ 上同调中在重言运算下封闭且包含重言子环的子环族，利用归纳论证证实了 Chenevier–Lannes 关于低次数上同调中伽罗瓦表示与霍奇结构的预测，并证明 $H^4$ 由重言类生成。
We bound from below the complexity of the top Chern class of the Hodge bundle in the Chow ring of the moduli space of curves: no formulas in terms of classes of degrees 1 and 2 can exist. As a consequence of the Torelli map, the 0-section over the second Voronoi compactification of the moduli of principally polarized abelian varieties also can not be expressed in terms of classes of degree 1 and 2. Along the way, we establish new cases of Pixton's conjecture for tautological relations. In the log Chow ring of the moduli space of curves, however, we prove that the top Chern class of the Hodge b…
Tautological classes of matroids
本文通过引入拟阵的重言类这一几何框架，统一并推广了拟阵理论与代数几何交叉领域的最新成果，建立了Tutte多项式四变量变换的Chow理论描述、对数凹性及例外Hirzebruch-Riemann-Roch型公式。
Let $Y$ be a smooth dimensionally transverse intersection of the Grassmannian $\hbox{Gr}(2,n)$ with 3 Plücker hyperplanes. We show that $Y$ admits a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of $Y$ injects into cohomology.
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 $n \leq 3g+5$.
The Beauville-Voisin conjecture for a hyperkähler manifold X states that the subring of the Chow ring A^*(X) generated by divisor classes and Chern characters of the tangent bundle injects into the cohomology ring of X. We prove a weak version of this conjecture when X is the Hilbert scheme of points on a K3 surface, for the subring generated by divisor classes and tautological classes. This in particular implies the weak splitting conjecture of Beauville for these geometries. In the process, we extend Lehn's formula and the Li-Qin-Wang W_{1+infinity} algebra action from cohomology to Chow gro…
The Chow rings of the moduli spaces of curves of genus 7, 8, and 9
本文确定了亏格7、8和9曲线模空间的有理Chow环是重言的，通过分析Hurwitz空间相交理论及特殊覆盖几何，并利用等变相交理论处理亏格9的六边形曲线轨迹。
The Chow Characteristic Image of \(\Spin(10)\) via the Affine Cone over the Spinor Variety
研究\(\Spin(10)\)的积分周环限制映射像，核心方法是构造类\(c_2c_3c_5\)，主要贡献是确定了模\(2\)和积分的周环特征标像，为相关研究提供关键结果。
Pandharipande-Pixton have used the geometry of the moduli space of stable quotients to produce relations between tautological Chow classes on the moduli space $M_g$ of smooth genus g curves. We study a natural extension of their methods to the boundary and more generally to Hassett's moduli spaces $\overline M_{g, w}$ of stable nodal curves with weighted marked points. Algebraic manipulation of these relations brings them into a Faber-Zagier type form. We show that they give Pixton's generalized FZ relations when all weights are one. As a special case, we give a formulation of FZ relations for…
We construct an action of the Neron--Severi part of the Looijenga-Lunts-Verbitsky Lie algebra on the Chow ring of the Hilbert scheme of points on a K3 surface. This yields a simplification of Maulik and Negut's proof that the cycle class map is injective on the subring generated by divisor classes as conjectured by Beauville. The key step in the construction is an explicit formula for Lefschetz duals in terms of Nakajima operators. Our results also lead to a formula for the monodromy action on Hilbert schemes in terms of Nakajima operators.
We give a new method of producing relations in the tautological ring R(M_{g, 1}), using the sl_2-action on the Chow ring of the universal Jacobian. With these relations, we prove that R(M_{g, 1}) is generated by κ_i for i no greater than g/3, together with ψ. Our computation shows that Faber's conjectures for M_{g, 1} are true for g up to 19. Further, by pushing relations forward to M_g, we obtain a new proof of Faber's conjectures (for M_g) for g up to 23. For g = 24, our method recovers all the Faber-Zagier relations. We also give an algebraic proof of an identity of Morita.
Rational Chow ring of the universal moduli space of semistable rank two bundles over genus two curves
本文确定亏格 2 曲线上秩 2 半稳定丛泛模空间的有理 Chow 环，证明其由重言类生成，并顺带得到相关泛 Jacobian 的 Chow 环。
We show that for a K3 surface X the finitely generated subring R(X) of the Chow ring introduced by Beauville and Voisin is preserved under derived equivalences. This is proved by analyzing Chern characters of spherical bundles. As for a K3 surface X defined over a number field all spherical bundles on the associated complex K3 surface are defined over $\bar\QQ$, this is compatible with the Bloch-Beilinson conjecture. Besides the work of Beauville and Voisin, Lazarfeld's result on Brill-Noether theory for curves in K3 surfaces and the deformation theory developed with Macri and Stellari in arXi…
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。
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 …
A smooth intersection $Y$ of two quadrics in $\mathbb{P}^{2g+1}$ has Hodge level 1. We show that such varieties $Y$ have a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of $Y$ injects into cohomology.
From $3$-spin to Negative $r$-spin: Limits of Cohomological Field Theories and Tautological Relations
本文证明负 r-自旋同调关系可由 Pixton 的 3-自旋关系推出，并由此推导其在同调 Chow 环中成立，特别地 r=2 时得到 Kazarian--Norbury K-关系在 Chow 环中成立。
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。
Weight decomposition for toroidal abelian fibrations
该研究探讨有理乘N映射在特定阿贝尔簇退化周环群上的作用，结合傅里叶变换，计算单位截面类、证明相对周动机广义权分解并确定重言环，属代数几何领域的相关研究。
Let M_{g,n}^rt be the moduli space of stable n-pointed curves of genus g>1 with rational tails. We also consider the space C_g^n classifying smooth curves of genus g with not necessarily distinct n ordered points. There is a natural proper map from M_{g,n}^rt to C_g^n which contracts all rational components. Tautological classes on these spaces are natural algebraic cycles reflecting the geometry of curves. In this short note we study the connection between tautological classes on M_{g,n}^rt and C_g^n. We show that there is a natural filtration on the tautological ring of M_{g,n}^rt consisting…
The rank-nullity ring of a matroid
本文提出 matroid 的秩-空环概念，并在均匀 matroid 情况下计算其 Hilbert 函数和 Gröbner 基。
We study the Chow ring of the boundary of the partial compactification of the universal family of principally polarized abelian varieties (ppav). We describe the subring generated by divisor classes, and compute the class of the partial compactification of the universal zero section, which turns out to lie in this subring. Our formula extends the results for the zero section of the universal uncompactified family. The partial compactification of the universal family of ppav can be thought of as the first two boundary strata in any toroidal compactification of the moduli space of ppav. Our form…
Zero cycles on the moduli space of curves
本研究针对曲线模空间上模点是否为tautological零闭链的问题，证明了有理曲面和K3曲面上曲线满足对应条件时模点为tautological的结论，建立了两类曲面相关tautological零闭链的联系。
We describe the Chow ring with rational coefficients of Mbar_{0,1}(P^n,d) as the subring of invariants of a ring B(Mbar_{0,1}(P^n,d);Q), relative to the action of the group of symmetries S_d. We compute B(Mbar_{0,1}(P^n,d);Q) by following a sequence of intermediate spaces for Mbar_{0,1}(P^n,d).
In this paper we construct and study the actions of certain deformations of the Lie algebra of Hamiltonians on the plane on the Chow groups (resp., cohomology) of the relative symmetric powers ${\cal C}^{[\bullet]}$ and the relative Jacobian ${\cal J}$ of a family of curves ${\cal C}/S$. As one of the applications, we show that in the case of a single curve $C$ this action induces a integral form of a Lefschetz $\operatorname{sl}_2$-action on the Chow groups of $C^{[N]}$. Another application gives a new grading on the ring of 0-cycles on the Jacobian $J$ of $C$ (with respect to the Pontryagin …
Let J be the Jacobian of a smooth curve C of genus g, and let A(J) be the ring of algebraic cycles modulo algebraic equivalence on J, tensored with Q. We study in this paper the smallest Q-vector subspace R of A(J) which contains C and is stable under the natural operations of A(J) : intersection and Pontryagin products, pull back and push down under multiplication by integers. We prove that this "tautological subring" is generated (over Q) by the classes of the subvarieties W_1=C, W_2=C+C, ..., W_{g-1}. If C admits a morphism of degree d onto P^1, we prove that the last d-1 classes suffice.
Non--tautological cycles on Prym moduli spaces
本文研究Prym模空间R_{g;m}的Chow环非tautology问题，通过bi-椭圆Prym曲线的组件R_B^0证明了特定类的非tautology，并扩展到紧凑模空间。
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。
Let $X$ a smooth quasi-projective algebraic surface, $L$ a line bundle on $X$. Let $X^{[n]}$ the Hilbert scheme of $n$ points on $X$ and $L^{[n]}$ the tautological bundle on $X^{[n]}$ naturally associated to the line bundle $L$ on $X$. We explicitely compute the image $\bkrh(L^{[n]})$ of the tautological bundle $L^{[n]}$ for the Bridgeland-King-Reid equivalence $\bkrh : \B{D}^b(X^{[n]}) \ra \B{D}^b_{\perm_n}(X^n)$ in terms of a complex $\comp{\mc{C}}_L$ of $\perm_n$-equivariant sheaves in $\B{D}^b_{\perm_n}(X^n)$. We give, moreover, a characterization of the image $\bkrh(L^{[n]} \tens ... \ten…
Let C be a curve over a non-singular base variety S. We study algebraic cycles on the symmetric powers C^[n] and on the Jacobian J. The Chow homology of C^[*], the sum of all C^[n], is a ring using the Pontryagin product. We prove that this ring is isomorphic to CH(J)[t]<u>, the PD-polynomial algebra (variable: u) over the usual polynomial ring (variable: t) over the Chow ring CH(J). We give two such isomorphisms that over a general base are different. Further we give some precise results on how CH(J) sits embedded in CH(C^[*]) and we give an explicit geometric description of how the derivatio…
The fifteenth cohomology group of moduli spaces of stable curves
本研究针对稳定曲线模空间的第十五上同调群问题，利用半重言扩张新框架，证明了特定情形下的相关结论，确定了对应向量空间的关系与显式基，并给出了若干新小对的结果。
We consider the variation of tropical Hodge structure (TVHS) associated to families of tropical varieties. The family of the tropical intermediate Jacobians of the associated tropical Hodge structure defines a bundle of tropical Jacobians, whose sections we call the tropical normal functions. We define formal sequential derivatives of these functions on the base with respect to the natural Gauss-Manin connection as the Hodge theoretic invariants detecting tropical cycles in the fibers. The associated invariants which are defined inductively are the higher Abel-Jacobi invariants in the tropical…
The eleventh cohomology group of $\bar{\mathcal{M}}_{g,n}$
本文证明 $\bar{\mathcal{M}}_{g,n}$ 的第十一有理上同调群几乎总为零，并利用 $H^{11}(\bar{\mathcal{M}}_{1,11})$ 构造出具有非零奇上同调和非重言闭链类的新模空间例子。
Let F be the complete flag variety over Spec(Z) with the tautological filtration 0 \subset E_1 \subset E_2 \subset ... \subset E_n=E of the trivial bundle E over F. The trivial hermitian metric on E(\C) induces metrics on the quotient line bundles L_i(\C). Let \hat{c}_1(L_i) be the first Chern class of L_i in the arithmetic Chow ring \hat{CH}(F) and x_i = -\hat{c}_1(L_i). Let h(X_1,...,X_n) be a polynomial with integral coefficients in the ideal <e_1,...,e_n> generated by the elementary symmetric polynomials e_i. We give an effective algorithm for computing the arithmetic intersection h(x_1,..…
We analyse the consequences of the Virasoro conjecture of Eguchi, Hori and Xiong for Gromov-Witten invariants, in the case of zero degree maps to the manifolds CP^1 and CP^2 (or more generally, smooth projective curves and smooth simply-connected projective surfaces). We obtain predictions involving intersections of psi and lambda classes on the compactification of M_{g,n}. In particular, we show that the Virasoro conjecture for CP^2 implies the numerical part of Faber's conjecture on the tautological Chow ring of M_g.
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