Moduli Space Of Stable Curves Research Articles

On the Chow and cohomology rings of moduli spaces of stable curves

In this paper, we ask: For which (g, n) is the rational Chow or cohomology ring of \bar{\mathcal{M}}_{g,n} generated by tautological classes? This question has been fully answered in genus 0 by Keel (the Chow and cohomology rings are tautological for all n (1992)) and genus 1 by Belorousski (the rings are tautological if and only if n \leq 10 (1998)). For g \geq 2 , work of van Zelm (2018) shows the Chow and cohomology rings are not tautological once 2g + n \geq 24 , leaving finitely many open cases. Here, we prove that the Chow and cohomology rings of \bar{\mathcal{M}}_{g,n} are isomorphic and generated by tautological classes for g = 2 and n \leq 9 and for 3 \leq g \leq 7 and 2g + n \leq 14 . For such (g, n) , this implies that the tautological ring is Gorenstein and \bar{\mathcal{M}}_{g,n} has polynomial point count.

A Generalization of Operads Based on Subgraph Contractions

Abstract We introduce a generalization of the notion of operad that we call a contractad, whose set of operations is indexed by connected graphs and whose composition rules are numbered by contractions of connected subgraphs. We show that many classical operads, such as the operad of commutative algebras, Lie algebras, associative algebras, pre-Lie algebras, the little disks operad, and the operad of moduli spaces of stable curves $\operatorname{\overline{{\mathcal{M}}}}_{0,n+1}$, admit generalizations to contractads. We explain that standard tools like Koszul duality and the machinery of Gröbner bases can be easily generalized to contractads. We verify the Koszul property of the commutative, Lie, associative, and Gerstenhaber contractads.

Polynomial point counts and odd cohomology vanishing on moduli spaces of stable curves

Polynomial point counts and odd cohomology vanishing on moduli spaces of stable curves

Holomorphic differential forms on moduli spaces of stable curves

We prove that the space of holomorphic p-forms on the moduli space overline{mathcal {M}}_{g,n} of stable curves of genus g with n marked points vanishes for p=14, 16, 18 unconditionally and also for p=20 under a natural assumption in the case g=3. This result is consistent with the Langlands program and it is obtained by applying the Arbarello–Cornalba inductive approach to the cohomology of moduli spaces.

An Algorithm to Compute the Fundamental Classes of Spin Components of Strata of Differentials

Abstract We construct an algorithm for computing the cycle classes of the spin components of a stratum of differentials in the moduli space of stable curves $\overline {\mathcal {M}}_{g,n}$. In addition, we implement it within the Sage package admcycles. Our main strategy is to reconstruct these cycles by their restrictions to the boundary of $\overline {\mathcal {M}}_{g,n}$ via clutching maps. These restrictions can be computed recursively by smaller dimensional spin classes and determine the original class via a certain system of linear equations. To study the spin parities on the boundary of a stratum of differentials of even type, we make use of the moduli space of multi-scale differentials introduced in [4]. As an application of our algorithm, one can verify a conjecture on spin double ramification cycles stated in [18] in many examples, by using the results computed by our algorithm.

Gromov–Witten theory of complete intersections via nodal invariants

AbstractWe provide an inductive algorithm computing Gromov–Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. We also prove that all Gromov–Witten classes of all smooth complete intersections in projective space belong to the tautological ring of the moduli space of stable curves. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov–Witten invariants, we introduce the new notion of nodal relative Gromov–Witten invariants. We then prove a nodal degeneration formula and a relative splitting formula. These results for nodal relative Gromov–Witten theory are stated in complete generality and are of independent interest.

The eleventh cohomology group of

AbstractWe prove that the rational cohomology group$H^{11}(\overline {\mathcal {M}}_{g,n})$vanishes unless$g = 1$and$n \geq 11$. We show furthermore that$H^k(\overline {\mathcal {M}}_{g,n})$is pure Hodge–Tate for all even$k \leq 12$and deduce that$\# \overline {\mathcal {M}}_{g,n}(\mathbb {F}_q)$is surprisingly well approximated by a polynomial inq. In addition, we use$H^{11}(\overline {\mathcal {M}}_{1,11})$and its image under Gysin push-forward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and nontautological algebraic cycle classes in Chow cohomology.

The quantum Witten–Kontsevich series and one-part double Hurwitz numbers

We study the quantum Witten-Kontsevich series introduced by Buryak, Dubrovin, Gu\'er\'e and Rossi in \cite{buryak2016integrable} as the logarithm of a quantum tau function for the quantum KdV hierarchy. This series depends on a genus parameter $\epsilon$ and a quantum parameter $\hbar$. When $\hbar=0$, this series restricts to the Witten-Kontsevich generating series for intersection numbers of psi classes on moduli spaces of stable curves. We establish a link between the $\epsilon=0$ part of the quantum Witten-Kontsevich series and one-part double Hurwitz numbers. These numbers count the number non-equivalent holomorphic maps from a Riemann surface of genus $g$ to $\mathbb{P}^{1}$ with a prescribe ramification profile over $0$, a complete ramification over $\infty$ and a given number of simple ramifications elsewhere. Goulden, Jackson and Vakil proved in \cite{goulden2005towards} that these numbers have the property to be polynomial in the orders of ramification over $0$. We prove that the coefficients of these polynomials are the coefficients of the quantum Witten-Kontsevich series. We also present some partial results about the full quantum Witten-Kontsevich power series.

Pseudostable Hodge Integrals

Abstract This paper initiates a study of Hodge integrals on moduli spaces of pseudostable curves. We prove an explicit comparison formula that allows one to effectively compute any pseudostable Hodge integral in terms of intersection numbers on moduli spaces of stable curves, and we use this comparison to prove that pseudostable Hodge integrals are equal to their stable counterparts when they are linear in lambda classes, but not when they are nonlinear. This suggests that pseudostable Gromov–Witten invariants are equal to usual Gromov–Witten invariants for target curves, but not for higher-dimensional target varieties.

A GENERALISATION OF WITTEN’S CONJECTURE FOR THE PIXTON CLASS AND THE NONCOMMUTATIVE KDV HIERARCHY

AbstractIn this paper, we formulate and present ample evidence towards the conjecture that the partition function (i.e. the exponential of the generating series of intersection numbers with monomials in psi classes) of the Pixton class on the moduli space of stable curves is the topological tau function of the noncommutative Korteweg-de Vries hierarchy, which we introduced in a previous work. The specialisation of this conjecture to the top degree part of Pixton’s class states that the partition function of the double ramification cycle is the tau function of the dispersionless limit of this hierarchy. In fact, we prove that this conjecture follows from the double ramification/Dubrovin–Zhang equivalence conjecture. We also provide several independent computational checks in support of it.

On the Goulden–Jackson–Vakil conjecture for double Hurwitz numbers

Goulden, Jackson and Vakil observed a polynomial structure underlying one-part double Hurwitz numbers, which enumerate branched covers of CP1 with prescribed ramification profile over ∞, a unique preimage over 0, and simple branching elsewhere. This led them to conjecture the existence of moduli spaces and tautological classes whose intersection theory produces an analogue of the celebrated ELSV formula for single Hurwitz numbers.In this paper, we present three formulas that express one-part double Hurwitz numbers as intersection numbers on certain moduli spaces. The first involves Hodge classes on moduli spaces of stable maps to classifying spaces; the second involves Chiodo classes on moduli spaces of r-spin curves; and the third involves tautological classes on moduli spaces of stable curves. We proceed to discuss the merits of these formulas against a list of desired properties enunciated by Goulden, Jackson and Vakil. Our formulas lead to non-trivial relations between tautological intersection numbers on moduli spaces of stable curves and hints at further structure underlying Chiodo classes. The paper concludes with generalisations of our results to the context of r-spin Hurwitz numbers.

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 available. In this paper we show how to apply the package, including concrete example computations.

The {mathcal {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 {mathcal {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 give additive generators for the {mathcal {H}}-tautological ring and show that their intersections may be algorithmically computed, building on work of Schmitt-van Zelm. As an application, we give a method for computing integrals of Harris-Mumford loci against tautological classes of complementary dimension, recovering and giving a mild generalization of a recent quasi-modularity result of the author for covers of elliptic curves.

Singular principal bundles on reducible nodal curves

Studying degenerations of moduli spaces of semistable principal bundles on smooth curves leads to the problem of studying moduli spaces on singular curves. In this note, we will see that the moduli spaces of δ \delta -semistable pseudo bundles on a nodal curve become, for large values of δ \delta , the moduli spaces of semistable singular principal bundles. The latter are reasonable candidates for degenerations and a potential basis for further developments as on irreducible nodal curves. In particular, we find a notion of semistability for principal bundles on reducible nodal curves. The understanding of the asymptotic behavior of δ \delta -semistability rests on a lemma from geometric invariant theory. The results allow for the construction of a universal moduli space of semistable singular principal bundles over the moduli space of stable curves. Due to recent work of Wilson, this universal moduli space has a close relation to the sheaf of algebras of conformal blocks.

Large genus asymptotics for intersection numbers and principal strata volumes of quadratic differentials

In this paper we analyze the large genus asymptotics for intersection numbers between $\psi$-classes, also called correlators, on the moduli space of stable curves. Our proofs proceed through a combinatorial analysis of the recursive relations (Virasoro constraints) that uniquely determine these correlators, together with a comparison between the coefficients in these relations with the jump probabilities of a certain asymmetric simple random walk. As an application of this result, we provide the large genus limits for Masur-Veech volumes and area Siegel-Veech constants associated with principal strata in the moduli space of quadratic differentials. These confirm predictions of Delecroix-Goujard-Zograf-Zorich from 2019.

Extended r-spin theory in all genera and the discrete KdV hierarchy

In this paper we construct a family of cohomology classes on the moduli space of stable curves generalizing Witten's r-spin classes. They are parameterized by a phase space which has one extra dimension and in genus 0 they correspond to the extended r-spin classes appearing in the computation of intersection numbers on the moduli space of open Riemann surfaces, while when restricted to the usual smaller phase space, they give in all genera the product of the top Hodge class by the r-spin class. They do not form a cohomological field theory, but a more general object which we call F-CohFT, since in genus 0 it corresponds to a flat F-manifold. For r=2 we prove that the partition function of such F-CohFT gives a solution of the discrete KdV hierarchy. Moreover the same integrable system also appears as its double ramification hierarchy.

Fibonacci, Golden Ratio, and Vector Bundles

There is a family of vector bundles over the moduli space of stable curves that, while first appearing in theoretical physics, has been an active topic of study for algebraic geometers since the 1990s. By computing the rank of the exceptional Lie algebra g2 case of these bundles in three different ways, a family of summation formulas for Fibonacci numbers in terms of the golden ratio is derived.

Quadratic double ramification integrals and the noncommutative KdV hierarchy

In this paper we compute the intersection number of two double ramification cycles (with different ramification profiles) and the top Chern class of the Hodge bundle on the moduli space of stable curves of any genus. These quadratic double ramification integrals are the main ingredient for the computation of the double ramification hierarchy associated to the infinite dimensional partial cohomological field theory given by $\exp(\mu^2 \Theta)$ where $\mu$ is a parameter and $\Theta$ is Hain's theta class, appearing in Hain's formula for the double ramification cycle on the moduli space of curves of compact type. This infinite rank double ramification hierarchy can be seen as a rank $1$ integrable system in two space and one time dimensions. We prove that it coincides with a natural analogue of the KdV hierarchy on a noncommutative Moyal torus.

A calculus for monomials in Chow group of zero cycles in the moduli space of stable curves

We introduce an algorithm for computing the value of all monomials in the Chow group of zero cycles in the moduli space of stable curves.

Counting Realizations of Laman Graphs on the Sphere

We present an algorithm that computes the number of realizations of a Laman graph on a sphere for a general choice of the angles between the vertices. The algorithm is based on the interpretation of such a realization as a point in the moduli space of stable curves of genus zero with marked points, and on the explicit description, due to Keel, of the Chow ring of this space.