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

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 $\bar{M}_{g,n}$

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.

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 $g$.

Relations in the tautological ring of $M_g$

Using a simple geometric argument, we obtain an infinite family of nontrivial relations in the tautological ring of $M_g$ (and in fact that of $M_{g,2}$). One immediate consequence of these relations is that the classes $κ_1,...,κ_{[g/3]}$ generate the tautological ring of $M_g$, which has been conjectured by Faber, and recently proven at the level of {\em cohomology} by Morita.

The tautological rings of the moduli spaces of stable maps

We show that the rational cohomology of the genus zero stable map spaces to SL flag varieties is entirely tautological.

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.

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。

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.

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.

Poincaré duality of wonderful compactifications and tautological rings

Let $g \geq 2$. Let $M_{g,n}^{rt}$ be the moduli space of $n$-pointed genus $g$ curves with rational tails. Let $C_g^n$ be the $n$-fold fibered power of the universal curve over $M_g$. We prove that the tautological ring of $M_{g,n}^{rt}$ has Poincaré duality if and only if the same holds for the tautological ring of $C_g^n$. We also obtain a presentation of the tautological ring of $M_{g,n}^{rt}$ as an algebra over the tautological ring of $C_g^n$. This proves a conjecture of Tavakol. Our results are valid in the more general setting of wonderful compactifications.

Gromov-Witten theory of target curves and the tautological ring

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.

A non-vanishing result for the tautological ring of {\cal M}_g

Looijenga recently proved that the tautological ring of M_g vanishes in degree d>g-2 and is at most one-dimensional in degree g-2, generated by the class of the hyperelliptic locus. Here we show that K_{g-2} is non-zero on M_g. The proof uses the Witten conjecture, proven by Kontsevich. With similar methods, we expect to be able to prove some (possibly all) of the identities in degree g-2 in the tautological ring that are part of the author's conjectural explicit description of the ring.

The Gorenstein conjecture fails for the tautological ring of $\mathcal{\bar M}_{2,n}$

Let $N$ be the smallest integer such that there is a non-tautological cohomology class of even degree on $\mathcal{\bar M}_{2,N}$. We remark that there is such a non-tautological class on $\mathcal{\bar M}_{2,20}$, by work of Graber and Pandharipande. We show that $\mathcal{\bar M}_{2,N}$ has non-tautological cohomology only in one degree, which is not the middle degree. In particular, it follows that the tautological ring of $\mathcal{\bar M}_{2,N}$ is not Gorenstein. We present some evidence suggesting that N=20 holds.

Relations in the Tautological Ring of the Universal Curve

We bound the dimensions of the graded pieces of the tautological ring of the universal curve from below for genus up to 27 and from above for genus up to 9. As a consequence we obtain the precise structure of the tautological ring of the universal curve for genus up to 9. In particular, we see that it is Gorenstein for these genera.

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…

$M_{25}$ 的 tautological 环不是 Gorenstein 的

The tautological ring of $M_{25}$ is not Gorenstein

本文证明了模空间 $M_{25}$ 的 tautological 环非 Gorenstein，推翻 Faber 猜想，并确认 Faber-Zagier 关系为完整关系列表。

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.…

On the tautological ring of Humbert curves

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.

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 …

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.

Cycles on curves and Jacobians: a tale of two tautological rings

We connect two notions of tautological ring: one for the moduli space of curves (after Mumford, Faber, etc.), and the other for the Jacobian of a curve (after Beauville, Polishchuk, etc.). The motivic Lefschetz decomposition on the Jacobian side produces relations between tautological classes, leading to results about Faber's Gorenstein conjecture on the curve side. We also relate certain Gorenstein properties on both sides and verify them for small genera. Further, we raise the question whether all tautological relations are motivic, giving a possible explanation why the Gorenstein properties…

On the tautological ring of a Jacobian modulo rational equivalence

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.

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.

On independence of generators of the tautological rings

We prove that all monomials of $κ$-classes and $ψ$-classes are independent in $R^k(\ocM_{g,n})/R^k(\partial\ocM_{g,n})$ for all $k \leq [g/3]$. We also give a simple argument for $κ_l \neq 0$ in $R^l(\mathcal{M}_g)$ for $l \leq g-2$.

Socle 配对在 tautological 环上的研究

Socle pairings on tautological rings

本文研究紧致型模空间上tautological环的λ_g配对，证明受限配对秩的公式及与λ_gλ_{g-1}配对秩的关系。

Powers of the theta divisor and relations in the tautological ring

We show that the vanishing of the $(g+1)$-st power of the theta divisor in the cohomology and Chow rings of the universal abelian variety implies, by pulling back along a collection of Abel-Jacobi maps, the vanishing results in the tautological ring of $\mathcal{M}_{g,n}$ of Looijenga, Ionel, Graber-Vakil, and Faber-Pandharipande. We also show that Pixton's double ramification cycle relations, which generalize the theta vanishing relations and were recently proved by the first and third authors, imply Theorem $\star$ of Graber and Vakil. Moreover, our proof provides an algorithm for expressing…

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。

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…

Calculating intersection numbers on moduli spaces of pointed curves

This is a note on calculating intersection numbers on moduli spaces of curves. A codimension 3 relation among tautological classes on the moduli space of genus 4 curves is given.

A vanishing result for tautological classes on the moduli of K3 surfaces

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.

Frobenius Manifolds near the Discriminant and Relations in the Tautological Ring

We give a criterion for extending a generically semisimple (not necessarily conformal) Frobenius manifold locally near a smooth point of the discriminant to a cohomological field theory. As an application, we show that a large set of tautological relations related to the Givental--Teleman classification for any generically semisimple cohomological field theories follow from Pixton's generalized Faber--Zagier relations.

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).

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…

Tautological rings of Hilbert modular varieties

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.

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…

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…

On the tautological rings of M_{g, 1} and its universal Jacobian

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.

$\overline{\mathcal{M}}_{g,n}$ 的环式环很少是 Gorenstein 环

The tautological ring of $\overline{\mathcal{M}}_{g,n}$ is rarely Gorenstein

本文证明模空间 $\overline{\mathcal{M}}_{g,n}$ 的环式环在 $g\geq 2$ 且 $2g+n\geq 24$ 时非 Gorenstein，推广了 Petersen 和 Tommasi 的结果，并猜想其逆命题成立，且证明了若干新情形。

Relations between tautological cycles on Jacobians

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…

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。

Chern Classes of Tautological Sheaves on Hilbert Schemes

We give an algorithmic description of the action of the Chern classes of tautological bundles on the cohomology of Hilbert schemes of points on surfaces within the framework of Nakajima's oscillator algebra. This leads to an identification of the cohomology ring of Hilbert schemes of the affine plane with a ring of differential operators on a Fock space. We end with the computation of the top Segre classes of tautological bundles associated to line bundles on Hilb^n up to n=7, and give a conjecture for the generating series.

Tautological ring of strata of differentials

Strata of $k$-differentials on smooth curves parameterize sections of the $k$-th power of the canonical bundle with prescribed orders of zeros and poles. Define the tautological ring of the projectivized strata using the $κ$ and $ψ$ classes of moduli spaces of pointed smooth curves along with the tautological class $η$ of the Hodge bundle. We show that if there is no pole of order $k$, then the tautological ring is generated by $η$ only, and otherwise it is generated by the $ψ$ classes corresponding to the poles of order $k$.

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。

On the tautological ring of $\M _g$

We prove among other things that any product of tautological classes of $\M_g$ of degree $d$ (in the Chow ring of $\M _g$ with rational coefficients) vanishes for $d>g-2$ and is proportional to the class of the hyperelliptic locus in degree $g-2$.

Invariance of tautological equations I: conjectures and applications

The main goal of this paper is to introduce a set of conjectures on the relations in the tautological rings. In particular, the framework gives an efficient algorithm to calculate all tautological equations using only finite dimensional linear algebra. Other applications are also indicated.

Tautological pairings on moduli spaces of curves

We discuss analogs of Faber's conjecture for two nested sequences of partial compactifications of the moduli space of smooth curves. We show that their tautological rings are one-dimensional in top degree but do not satisfy Poincare duality.

Tautological and non-tautological cohomology of the moduli space of curves

After a short exposition of the basic properties of the tautological ring of the moduli space of genus g Deligne-Mumford stable curves with n markings, we explain three methods of detecting non-tautological classes in cohomology. The first is via curve counting over finite fields. The second is by obtaining length bounds on the action of the symmetric group S_n on tautological classes. The third is via classical boundary geometry. Several new non-tautological classes are found.

论稳定曲线模空间的 Chow 环与上同调环

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。

Tautological classes on low-degree Hurwitz spaces

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 $k \leq 5$.

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.

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…

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.

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.

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 …