1999年，van der Geer定义并计算了主极化阿贝尔簇（维数为$g$）的模空间$\mathcal{A}_g$的重言Chow环。通过研究从$\mathcal{A}_g$到紧致型亏格$g$曲线模空间$\mathcal{M}_g^{\mathrm{ct}}$的Torelli拉回代数环类，我们证明了乘积类$[\mathcal{A}_1\times \mathcal{A}_5]\in \mathsf{CH}^{5}(\mathcal{A}_6)$是非重言的，这是阿贝尔簇模空间上第一个有趣的非重言代数类的构造。我们的证明使用了Pixton猜想并由Canning-Larson-Schmitt最近证明的亏格6紧致型模空间$\mathcal{M}_6^{\mathrm{ct}}$的重言环$\mathsf{R}^*(\mathcal{M}_6^{\mathrm{ct}})$的完整描述。重言环$\mathsf{R}^*(\mathcal{M}_6^{\mathrm{ct}})$具有一维Gorenstein核，该核由$[\mathcal{A}_1\times \mathcal{A}_5]$的Torelli拉回给出几何解释。更一般地，$[\mathcal{A}_1\times \mathcal{A}_{g-1}]$与其重言投影之差的Torelli拉回总是位于$\mathsf{R}^*(\mathcal{M}_g^{\mathrm{ct}})$的Gorenstein核中。乘积映射$\mathcal{A}_1\times \mathcal{A}_{g-1}\rightarrow \mathcal{A}_g$是具有一般Neron-Severi秩2的Noether-Lefschetz轨迹。通过包含更一般的Noether-Lefschetz轨迹，可以自然扩展van der Geer的重言环。我们给出了与所有$g$的Noether-Lefschetz轨迹的环类相关的结果和猜想。
英文摘要：
The tautological Chow ring of the moduli space $\mathcal{A}_g$ of principally polarized abelian varieties of dimension $g$ was defined and calculated by van der Geer in 1999. By studying the Torelli pullback of algebraic cycles classes from $\mathcal{A}_g$ to the moduli space $\mathcal{M}_g^{\mathrm{ct}}$ of genus $g$ of curves of compact type, we prove that the product class $[\mathcal{A}_1\times \mathcal{A}_5]\in \mathsf{CH}^{5}(\mathcal{A}_6)$ is non-tautological, the first construction of an interesting non-tautological algebraic class on the moduli spaces of abelian varieties. For our proof, we use the complete description of the the tautological ring $\mathsf{R}^*(\mathcal{M}_6^{\mathrm{ct}})$ in genus 6 conjectured by Pixton and recently proven by Canning-Larson-Schmitt. The tautological ring $\mathsf{R}^*(\mathcal{M}_6^{\mathrm{ct}})$ has a 1-dimensional Gorenstein kernel, which is geometrically explained by the Torelli pullback of $[\mathcal{A}_1\times \mathcal{A}_5]$. More generally, the Torelli pullback of the difference between $[\mathcal{A}_1\times \mathcal{A}_{g-1}]$ and its tautological projection always lies in the Gorenstein kernel of $\mathsf{R}^*(\mathcal{M}_g^{\mathrm{ct}})$.
The product map $\mathcal{A}_1\times \mathcal{A}_{g-1}\rightarrow \mathcal{A}_g$ is a Noether-Lefschetz locus with general Neron-Severi rank 2. A natural extension of van der Geer's tautological ring is obtained by including more general Noether-Lefschetz loci. Results and conjectures related to cycle classes of Noether-Lefschetz loci for all $g$ are presented.
Looijenga's vanishing theorem on the moduli space of curves $M_g$ says that the tautological ring vanishes above degree $g-2$. We prove an analogous result for the tautological cohomology ring of the moduli space of K3 surfaces.
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…
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.
Let $J$ be the Jacobian of a smooth projective complex curve $C$ which admits non-trivial automorphisms, and let $A(J)$ be the ring of algebraic cycles on $J$ with rational coefficients modulo algebraic equivalence. We present new tautological rings in $A(J)$ which extend in a natural way the tautological ring studied by Beauville (Compos Math 140(3):683-688, 2004). We then show there exist tautological rings induced on special complementary abelian subvarieties of $J$.
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.
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.
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…
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…
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).
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 compute the tautological ring for a Hilbert modular variety at an unramified prime. The method generalises that of van der Geer from the Siegel case.
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.
Tautological systems developed in [6],[7] are Picard-Fuchs type systems to study period integrals of complete intersections in Fano varieties. We generalize tautological systems to local complete intersections, which are zero loci of global sections of vector bundles over Fano varieties. In particular, we obtain similar criterion as [6, 7] about holonomicity and regularity of the system. We also prove solution rank formulas and geometric realizations of solutions following the work of hypersurfaces [5, 4].
Logarithmic series and Hodge integrals in the tautological ring (with an appendix by D. Zagier)
我们描述了一个新的角度上的相交理论的模空间的曲线涉及Virasoro约束和Gorenstein条件。本文的主要结果是计算了非奇异曲线模空间重言式环上的一个基本1点Hodge积分级数。在附录D. Zagier，“Polynomials arising from the tautological ring”，详细研究了模空间上系数为相交数的多项式。本文和附录一起证明了重言式环上所有已证明的1-点积分公式，以及这些公式的自然扩张。
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.…
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.
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.
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.
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.
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…
In the Gromov-Witten theory of a target curve we consider descendent integrals against the virtual fundamental class relative to the forgetful morphism to the moduli space of curves. We show that cohomology classes obtained in this way lie in the tautological ring.
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…
