Note on tautological classes of moduli of K3 surfaces

Tautological classes on the moduli spaces of stable maps to projective spaces

We present a localization proof of the fact that the cohomology of the moduli spaces of genus zero stable maps to projective spaces is entirely tautological. In addition, we obtain a description of a Bialynicki-Birula stratification in the context of Gathmann's moduli spaces of maps with prescribed contact order to a fixed hyperplane.

Tautological classes on moduli space of hyperkähler manifolds

In this paper, we discuss the cycle theory on moduli spaces $\cF_h$ of $h$-polarized hyperkähler manifolds. Firstly, we construct the tautological ring on $\cF_h$ following the work of Marian, Oprea and Pandharipande on the tautological conjecture on moduli spaces of K3 surfaces. We study the tautological classes in cohomology groups and prove that most of them are linear combinations of Noether-Lefschetz cycle classes. In particular, we prove the cohomological version of the tautological conjecture on moduli space of K3$^{[n]}$-type hyperkähler manifolds with $n\leq 2$. Secondly, we prove the…

Complete descriptions of the tautological rings of the moduli spaces of curves of genus lower or equal to 4 with marked points

We study here the tautological rings of the moduli spaces of compact Riemann surfaces of genus 1,2,3 and 4 with marked points. The paper presents the complete descriptions of these rings by describing the groups of all degrees.

The tautological ring of the moduli space M_{2,n}^rt

We study the tautological ring of the moduli space of stable n-pointed curves of genus two with rational tails. The algebra is described in terms of explicit generators and relations. It is proven that this algebra is Gorenstein.

Tautological bundles on parabolic moduli spaces: Euler characteristics and Hecke correspondences

We calculate the Euler characteristic of associated vector bundles over the moduli spaces of stable parabolic bundles on smooth curves. Our method is based on a wall-crossing technique from Geometric Invariant Theory, certain iterated residue calculus and the tautological Hecke correspondence. Our work was motivated by the results of Teleman and Woodward on the index of K-theory classes on moduli stacks.

Tautological classes on the moduli space of hyperelliptic curves with rational tails

We study tautological classes on the moduli space of stable $n$-pointed hyperelliptic curves of genus $g$ with rational tails. Our result gives a complete description of tautological relations. The method is based on the approach of Yin in comparing tautological classes on the moduli of curves and the universal Jacobian. It is proven that all relations come from the Jacobian side. The intersection pairings are shown to be perfect in all degrees. We show that the tautological algebra coincides with its image in cohomology via the cycle class map. The latter is identified with monodromy invarian…

Hilbert 概形上广义重言丛的模空间

Moduli spaces of generalised tautological bundles on Hilbert schemes

本文在代数曲面点的 Hilbert 概形上构造了由模空间连通分支参数化的新稳定向量丛，推广了先前关于重言丛及 O'Grady 的近期研究。

Tautological algebra of the moduli space of semistable bundles on an elliptic curve

In this paper, our aim is to find the relations amongst the cohomology classes of Brill-Noether subvarieties of the moduli space of semistable bundles over an elliptic curve. We obtain results similar to the Poincaré relations on a Jacobian variety.

Relative maps and tautological classes

The moduli space of stable relative maps to the projective line combines features of stable maps and admissible covers. We prove all standard Gromov-Witten classes on these moduli spaces of stable relative maps have tautological push-forwards to the moduli space of curves. In particular, the fundamental classes of all moduli spaces of admissible covers push-forward to tautological classes. Consequences for the tautological rings of the moduli spaces of curves include methods for generating new relations, uniform derivations of the socle and vanishing statements of the Gorenstein conjectures fo…

Tautological classes on moduli spaces of curves with linear series and a push-forward formula when $ρ=0$

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.

Relations in the tautological ring of the moduli space of K3 surfaces

We study the interplay of the moduli of curves and the moduli of K3 surfaces via the virtual class of the moduli spaces of stable maps. Using Getzler's relation in genus 1, we construct a universal decomposition of the diagonal in Chow in the third fiber product of the universal K3 surface. The decomposition has terms supported on Noether-Lefschetz loci which are not visible in the Beauville-Voisin decomposition for a fixed K3 surface. As a result of our universal decomposition, we prove the conjecture of Marian-Oprea-Pandharipande: the full tautological ring of the moduli space of K3 surfaces…

admcycles -- a Sage package for calculations in the tautological ring of the moduli space of stable curves

The tautological ring of the moduli space of stable curves has been studied extensively in the last decades. We present a SageMath implementation of many core features of this ring. This includes lists of generators and their products, intersection numbers and verification of tautological relations. Maps between tautological rings induced by functoriality, that is pushforwards and pullbacks under gluing and forgetful maps, are implemented. Furthermore, many interesting cycle classes, such as the double ramification cycles, strata of k-differentials and hyperelliptic or bielliptic cycles are av…

Relations in the tautological ring of the moduli space of curves

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).

光滑带点曲线的模空间上的非上同调环

Non-tautological cycles on moduli spaces of smooth pointed curves

本研究推广了已有方法，证明除有限多组亏格$g$和点数$n$外，光滑带点曲线模空间$\bM_{g,n}$上均存在非上同调代数上同调类，覆盖了绝大多数此前未解决的情形。

Relative virtual localization and vanishing of tautological classes on moduli spaces of curves

We prove a localization formula for the moduli space of stable relative maps. As an application, we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i-g+1 genus 0 components. (The proof of this result is quite short and naive.) As consequences, we prove and generalize various conjectures and theorems about various moduli spaces of curves (due to Getzler, Ionel, Faber, Looijenga, Pandharipande, Diaz, and others). This theorem appears to be the geometric content behind these results; the r…

Tautological ring of the moduli space of generalised parabolic line bundles on a curve

In this paper, we consider the tautological ring containing the extended Brill-Noether algebraic classes on the normalization of the compactified Jacobian of a complex nodal projective curve (with one node). This smallest $\Q$-subalgebra of algebraic classes under algebraic equivalence, stable under extensions of the maps induced by multiplication maps, Pontrayagin product and Fourier transform, is shown to be generated by pullback of the Brill-Noether classes of the Jacobian of the normalized curve and some natural classes.

曲线模空间的相交理论导论

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。

Tautological relations in moduli spaces of weighted pointed curves

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…

On the tautological ring of $\bar{M}_{g,n}$

We prove that the dimension 0 part of the tautological ring of the moduli space of stable pointed curves is one-dimensional. This provides the first genus-free evidence for a conjecture of Faber and Pandharipande that the tautological ring of the moduli space is Gorenstein, answering in the affirmative a question of Hain and Looijenga.

Tautological relations for stable maps to a target variety

We define tautological relations for the moduli space of stable maps to a target variety. Using the double ramification cycle formula for target varieties of Janda-Pandharipande-Pixton-Zvonkine, we construct nontrivial tautological relations parallel to Pixton's double ramification cycle relations for the moduli of curves. Examples and applications are discussed.

曲线模空间上的全纯形式与非重言闭链

Holomorphic forms and non-tautological cycles on moduli spaces of curves

本文利用曲线模空间上的全纯形式构造新的非重言代数上同调类，推广了Graber–Pandharipande和van Zelm关于双椭圆闭链的工作，证明了无穷多个$\mathcal{M}_{g,n}$及$g=12$和$g\geq16$时$\mathcal{M}_g$上存在非重言类。

曲线模空间的对数重言环

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。

正合微分的分层类

Tautological classes of strata of exact differentials

本文研究正合微分分层，证明其闭包类位于多尺度微分模空间的除子型示性环内，并借助有理函数关联给出了Hurwitz空间类示性的新证明与计算方法。

曲线模空间上的同调形式环

Rings of tautological forms on moduli spaces of curves

本文定义并研究了带标曲线模空间上微分形式的同调形式环系统，证明了其有限维性，并刻画了Kawazumi-Zhang不变量为唯一满足特定同调条件的平滑函数。

Logarithmic series and Hodge integrals in the tautological ring (with an appendix by D. Zagier)

We describe a new perspective on the intersection theory of the moduli space of curves involving both Virasoro constraints and Gorenstein conditions. The main result of the paper is the computation of a basic 1-point Hodge integral series occurring in the tautological ring of the moduli space of nonsingular curves. In the appendix by D. Zagier, "Polynomials arising from the tautological ring", a detailed study is made of certain polynomials whose coefficients are intersection numbers on moduli space. The paper and the appendix together provide proofs of all previously conjectured formulas for …

The H-tautological ring

We extend the theory of tautological classes on moduli spaces of stable curves to the more general setting of moduli spaces of admissible Galois covers of curves, introducing the so-called H-tautological ring. The main new feature is the existence of restriction-corestriction morphisms remembering intermediate quotients of Galois covers, which are a rich source of new classes. In particular, our new framework includes classes of Harris-Mumford admissible covers on moduli spaces of curves, which are known in some (and speculatively many more) examples to lie outside the usual tautological ring.…

阿贝尔簇模空间上的重言与非重言环类

Tautological and non-tautological cycles on the moduli space of abelian varieties

本文通过Torelli拉回研究阿贝尔簇模空间的重言环，证明$[\mathcal{A}_1\times \mathcal{A}_5]$在$\mathcal{A}_6$中非重言，并利用Gorenstein核给出几何解释，同时探讨Noether-Lefschetz轨迹的推广。

Relations in the tautological ring

These notes cover our series of three lectures at Humboldt University in Berlin for the October 2010 conference "Intersection theory on moduli space" (organized by G. Farkas). The topic concerns relations among the kappa classes in the tautological ring of the moduli space of genus g curves. After a discussion of classical constructions in Wick form, we derive an explicit set of relations obtained from the virtual geometry of the moduli space of stable quotients. In a series of steps, the stable quotient relations are transformed to simpler and simpler forms. Our final result establishes a pre…

Conjectural relations in the tautological ring of $\bar{M}_{g,n}$

We describe a very large class of conjectural relations in the tautological ring of the moduli space $\bar{M}_{g,n}$ of stable curves of genus $g$ with $n$ marked points, extending and generalizing the Faber-Zagier relations. These notes are loosely based on informal talks given by the author at the workshop at KTH Stockholm on "The moduli space of curves and its intersection theory" in April 2012.

Deligne的重量谱序列与曲线模空间的塔罗格(cohomology)

Deligne's weight spectral sequence and tautological cohomology of the moduli space of curves

本文通过计算机程序实现Deligne的重量谱序列，计算标号光滑曲线模空间的有理上同调的重量分级，特别针对 genus 五和 genus 三的情况，研究了边界分层中上同调群由塔罗格类生成时的结构。

曲线模空间上的零闭链

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。

Weierstrass cycles and tautological rings in various moduli spaces of algebraic curves

We analyze Weierstrass cycles and tautological rings in moduli space of smooth algebraic curves and in moduli spaces of integral algebraic curves with embedded disks with special attention to moduli spaces of curves having genus $\leq 6$. In particular, we show that our general formula gives a good estimate for the dimension of Weierstrass cycles for lower genera.

Intersections of Tautological Classes on Blowups of Moduli Spaces of Genus-One Curves

We give two recursions for computing top intersections of tautological classes on blowups of moduli spaces of genus-one curves. One of these recursions is analogous to the well-known string equation. As shown in previous papers, these numbers are useful for computing genus-one enumerative invariants of projective spaces and Gromov-Witten invariants of complete intersections.

Cohomological field theories with non-tautological classes

A method of constructing Cohomological Field Theories (CohFTs) with unit using minimal classes on the moduli spaces of curves is developed. As a simple consequence, CohFTs with unit are found which take values outside of the tautological cohomology of the moduli spaces of curves. A study of minimal classes in low genus is presented in the Appendix by D. Petersen.

阿贝尔簇模空间上闭链的 tautological 投影

Tautological projection for cycles on the moduli space of abelian varieties

本文定义了主极化阿贝尔簇模空间上代数闭链类的tautological投影算子，通过证明环面紧化边界上霍奇丛最高陈类为零完成定义，还计算了乘积闭链投影、讨论相关计算并提出开放问题与猜想。

Combinatorial classes on the moduli space of curves are tautological

The combinatorial description via ribbon graphs of the moduli space of Riemann surfaces makes it possible to define combinatorial cycles in a natural way. Witten and Kontsevich first conjectured that these classes are polynomials in the tautological classes. We answer affirmatively to this conjecture and find recursively all the polynomials.

The tautological ring of $M_{1,n}^{ct}$

We describe the tautological ring of the moduli space of $n$-pointed curves of genus one of compact type. It is proven that it is a Gorenstein algebra.

Chow Rings of Fine Quiver Moduli are Tautologically Presented

A result of A. King and C. Walter asserts that the Chow ring of a fine quiver moduli space is generated by the Chern classes of universal bundles if the quiver is acyclic. We will show that defining relations between these Chern classes arise geometrically as degeneracy loci associated to the universal representation.

Computing top intersections in the tautological ring of $M_g$

We derive effective recursion formulae of top intersections in the tautological ring $R^*(M_g)$ of the moduli space of curves of genus $g\geq 2$. As an application, we prove a convolution-type tautological relation in $R^{g-2}(M_g)$.

On the tautological ring of $M_{g,n}$

We state and check the analogue of Faber's conjectures for the tautological ring of the moduli spaces $M_{g,n}$ of $n$-pointed smooth curves of genus $g$.

关于扭曲线模栈的 tautological 环

On the tautological rings of stacks of twisted curves

该研究引入扭曲线模栈的tautological环，同时建立了这类环的一些基本性质，为相关代数几何研究提供新的研究对象与基础性质支撑。

Generators for the cohomology ring of the moduli of one-dimensional sheaves on $\mathbb{P}^2$

We explore the structure of the cohomology ring of the moduli space of stable 1-dimensional sheaves on $\mathbb{P}^2$ of any degree. We obtain a minimal set of tautological generators, which implies an optimal generation result for both the cohomology and the Chow ring of the moduli space. Our approach is through a geometric study of tautological relations.

The fourth tautological group of $\bar{\mathcal{M}_{g,n}}$ and relations with the cohomology

We give a complete description of the tautological subgroup of the fourth cohomology group of the moduli space of pointed stable curves, and prove that for g \geq 8 it coincides with the cohomology group itself. We further give a conjectural upper bound depending on the genus for the degree of new tautological relations.

Extension groups of Tautological Bundles on Symmetric Products of Curves

We provide a spectral sequence computing the extension groups of tautological bundles on symmetric products of curves. One main consequence is that, if $E\neq \mathcal O_X$ is simple, then the natural map $\operatorname*{Ext}^1(E,E)\to \operatorname*{Ext}^1(E^{[n]},E^{[n]})$ is injective for every $n$. Along with previous results, this implies that $E\mapsto E^{[n]}$ defines an embedding of the moduli space of stable bundles of slope $μ\notin[-1,n-1]$ on the curve $X$ into the moduli space of stable bundles on the symmetric product $X^{(n)}$. The image of this embedding is, in most cases, cont…

Top tautological group of $\mathcal M_{g,n}$

We describe the structure of the top tautological group in the cohomology of the moduli space of smooth genus $g$ curves with $n$ marked points.

Tautological relations via r-spin structures

Relations among tautological classes on the moduli space of stable curves are obtained via the study of Witten's r-spin theory for higher r. In order to calculate the quantum product, a new formula relating the r-spin correlators in genus 0 to the representation theory of sl2 is proven. The Givental-Teleman classification of CohFTs is used at two special semisimple points of the associated Frobenius manifold. At the first semisimple point, the R-matrix is exactly solved in terms of hypergeometric series. As a result, an explicit formula for Witten's r-spin class is obtained (along with tautolo…

Tautological rings of stable map spaces

We find a set of generators and relations for the system of extended tautological rings associated to the moduli spaces of stable maps in genus zero, admitting a simple geometrical interpretation. In particular, when the target is $\PP^n$, these give a complete presentation for the cohomology and Chow rings in the cases with/without marked points.

紧致型模空间的Gorenstein性质与Pixton猜想

The Gorenstein property and Pixton's conjecture for compact type moduli

该研究证明了紧致型模空间$\u039c_{g,n}^{\mathrm{ct}}$在特定参数下重言环非Gorenstein，并验证了Pixton猜想的多个新案例，相关结果是主极化阿贝尔簇模空间非重言闭链研究的关键基础。

Descendent integrals and tautological rings of moduli spaces of curves

The main objective of this paper is to give a summary of our recent work on recursion formulae for intersection numbers on moduli spaces of curves and their applications. We also present a conjectural relation between tautological rings and the mock theta function.

Polynomial families of tautological classes on $\mathcal{M}_{g,n}^{rt}$

We study classes $P_{g,T}(α;β)$ on the moduli space of stable, genus g curves with rational tails defined by pushing forward the virtual fundamental classes of spaces of relative stable maps to an unparameterized projective line. A comparison with classes $Q_{g,T}$ arising from sections of the universal Jacobian shows the classes $P_{g,T}(α;β)$ are polynomial in the parts of the partitions indexing the special ramification data. Virtual localization on moduli spaces of relative stable maps gives sufficient relations to compute the coefficients of these polynomials in various cases.

Constructions of nontautological classes on moduli spaces of curves

We construct explicit examples of algebraic cycles in \bar M_g (for large g congruent to 2 mod 4) and in M_2,20 (no bar) which are not in the tautological ring. In an appendix we give a general method for computing intersections in the tautological ring.

The tautological ring of $\mathcal{M}_{g,n}$ via Pandharipande-Pixton-Zvonkine $r$-spin relations

We use relations in the tautological ring of the moduli spaces $\overline{\mathcal{M}}_{g,n}$ derived by Pandharipande, Pixton, and Zvonkine from the Givental formula for the $r$-spin Witten class in order to obtain some restrictions on the dimensions of the tautological rings of the open moduli spaces $\mathcal{M}_{g,n}$. In particular, we give a new proof for the result of Looijenga (for $n=1$) and Buryak et al. (for $n\geq 2$) that $\dim R^{g-1}(\mathcal{M}_{g,n}) \leq n$. We also give a new proof of the result of Looijenga (for $n=1$) and Ionel (for arbitrary $n\geq 1$) that $R^{i}(\mathca…

拟射影堆栈上秩2稳定丛的拟代数

Tautological algebra of the moduli stack of semistable bundles of rank 2 on a general curve

研究一般曲线上的秩2稳定丛模堆栈的拟代数结构，揭示其由除子类生成的性质，扩展经典Porteous公式的应用。

A calculus for the moduli space of curves

This article accompanies my lecture at the 2015 AMS summer institute in algebraic geometry in Salt Lake City. I survey the recent advances in the study of tautological classes on the moduli spaces of curves. After discussing the Faber-Zagier relations on the moduli spaces of nonsingular curves and the kappa rings of the moduli spaces of curves of compact type, I present Pixton's proposal for a complete calculus of tautological classes on the moduli spaces of stable curves. Several open questions are discussed. An effort has been made to condense a great deal of mathematics into as few pages as…