In this paper we analyze the large genus asymptotics for intersection numbers between $ψ$-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.
In this paper we consider the large genus asymptotics for Masur-Veech volumes of arbitrary strata of Abelian differentials. Through a combinatorial analysis of an algorithm proposed in 2002 by Eskin-Okounkov to exactly evaluate these quantities, we show that the volume $ν_1 \big( \mathcal{H}_1 (m) \big)$ of a stratum indexed by a partition $m = (m_1, m_2, \ldots , m_n)$ is $\big( 4 + o(1) \big) \prod_{i = 1}^n (m_i + 1)^{-1}$ as $2g - 2 = \sum_{i = 1}^n m_i$ tends to $\infty$. This confirms a prediction of Eskin-Zorich and generalizes some of the recent results of Chen-Moeller-Zagier and Sauva…
We state conjectures on the asymptotic behavior of the Masur-Veech volumes of strata in the moduli spaces of meromorphic quadratic differentials and on the asymptotics of their area Siegel-Veech constants as the genus tends to infinity.
We address the question concerning the birational geometry of the strata of holomorphic and quadratic differentials. We show strata of holomorphic and quadratic differentials to be uniruled in small genus by constructing rational curves via pencils on K3 and del Pezzo surfaces respectively. Restricting to genus $3\leq g\leq6$, we construct projective bundles over a rational varieties that dominate the holomorphic strata with length at most $g-1$, hence showing in addition that these strata are unirational.
We give a description of the Kontsevich-Zorich strata for genus 3 in terms of root system data. For each non-open stratum we obtain a presentation of its orbifold fundamental group.
We give an algebraic way of distinguishing the components of the exceptional strata of quadratic differentials in genus three and four. The complete list of these strata is (9, -1), (6,3,-1), (3,3,3, -1) in genus three and (12), (9,3), (6,6), (6,3,3) and (3,3,3,3) in genus four. This result is part of a more general investigation of disjointness of Teichmueller curves with divisors of Brill-Noether type on the moduli space of curves. As a result we show that for many strata of quadratic differentials in low genus the sum of Lyapunov exponents for the Teichmueller geodesic flow is the same for …
We consider two applications of the strata of differentials of the second kind (all residues equal to zero) with fixed multiplicities of zeros and poles: Positivity: In genus $g=0$ we show any associated divisorial projection to $\overline{\mathcal{M}}_{0,n}$ is $F$-nef and hence conjectured to be nef. We compute the class for all genus when the divisorial projection forgets only simple zeros and show in these cases the genus $g=0$ projections are indeed nef. Hurwitz spaces: We show the Hurwitz spaces of degree $d$, genus $g$ covers of $\mathbb{P}^1$ with pure branching (one ramified point ove…
We describe the invariants of plane quartic curves -- nonhyperelliptic genus 3 curves in their canonical model -- as determined by Dixmier and Ohno, with application to the classification of curves with given structure. In particular, we determine modular equations for the strata in the moduli space $\Me_3$ of plane quartics which have at least seven hyperflexes, and obtain an computational characterization of curves in these strata.
We prove that for a generic element in a nonhyperelliptic component of an abelian stratum $\mathcal{H}_g(μ)$ in genus $g$, the underlying curve has maximal gonality. We extend this result to the case of quadratic strata when the partition $μ$ has positive entries. As a consequence we deduce that all nonhyperelliptic components of $\mathcal{H}_9(μ)$ are uniruled when $μ$ is a positive partition of 16 and all nonhyperelliptic components of $\mathcal{H}^2_g(μ)$ are uniruled when $μ$ is a positive partition of $4g-4$ and either $3\leq g\leq5$ or $g=6$ and $l(μ)\geq 4$.
We study the induced Ekedahl-Oort stratification on the moduli space $\mathcal{M}_4$ of curves of genus $4$ in characteristic $3$. By constructing families of curves with given Ekedahl-Oort type and by computing the dimension of the intersection of the induced Ekedahl-Oort strata and the boundary divisor classes, we show that for certain induced Ekedahl-Oort strata in $\mathcal{M}_4$, they have the same codimension in $\mathcal{M}_4$ as the corresponding Ekedahl-Oort strata in moduli space $\mathcal{A}_4$ of principally polarized abelian varieties of dimension $4$.
Non-asphericity of strata of genus-one differentials and stability spaces
Boston College（波士顿学院）；The Ohio State University（俄亥俄州立大学）；Tsinghua University（清华大学）；Beijing Institute of Mathematical Sciences and Applications（北京数学科学与应用研究院）；Zhejiang University（浙江大学）
We show that the strata $ \mathcal{M}_{11,6}(k) \subset \mathcal{M}_{11} $ of $ 6-$gonal curves of genus $ 11 $, equipped with $k$ mutually independent and type I pencils of degree six, have a unirational irreducible component for $5\leq k\leq 9$. The unirational families arise from degree $ 9 $ plane curves with $ 4 $ ordinary triple and $ 5 $ ordinary double points that dominate an irreducible component of expected dimension. We will further show that the family of degree $ 8 $ plane curves with $ 10 $ ordinary double points covers an irreducible component of excess dimension in $ \mathcal{M…
This paper focuses on the interplay between the intersection theory and the Teichmueller dynamics on the moduli space of curves. As applications, we study the cycle class of strata of the Hodge bundle, present an algebraic method to calculate the class of the divisor parameterizing abelian differentials with a non-simple zero, and verify a number of extremal effective divisors on the moduli space of pointed curves in low genus.
Algebraic varieties and curves arising in Birkhoff strata of the Sato Grassmannian Gr${^{(2)}}$ are studied. It is shown that the big cell $Σ_0$ contains the tower of families of the normal rational curves of all odd orders. Strata $Σ_{2n}$, $n=1,2,3,...$ contain hyperelliptic curves of genus $n$ and their coordinate rings. Strata $Σ_{2n+1}$, $n=0,1,2,3,...$ contain $(2m+1,2m+3)-$plane curves for $n=2m,2m-1$ $(m \geq 2)$ and $(3,4)$ and $(3,5)$ curves in $Σ_3$, $Σ_5$ respectively. Curves in the strata $Σ_{2n+1}$ have zero genus.
We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p > 2. Using this, we prove that the Z/\ell-monodromy of every irreducible component of the stratum H_g^f of hyperelliptic curves of genus g and p-rank f is the symplectic group Sp_{2g}(Z/\ell) if g > 2, f > 0 and \ell is an odd prime distinct from p. These results yield applications about the generic behavior of hyperelliptic curves of given genus and p-rank. The first application is that a generic hyperelliptic curve of genus g > 2 and p-rank 0 is not su…
Affine varieties among all algebraic varieties have simple structures. For example, an affine variety does not contain any complete algebraic curve. In this paper we study affine related properties of strata of $k$-differentials on smooth curves which parameterize sections of the $k$-th power of the canonical line bundle with prescribed orders of zeros and poles. We show that if there is a prescribed pole of order at least $k$, then the corresponding stratum does not contain any complete curve. Moreover, we explore the amusing question whether affine invariant manifolds arising from Teichmülle…
This paper is devoted to the classification of connected components of Prym eigenform loci in the strata H(2,2)^odd and H(1,1,2) in the Abelian differentials bundle in genus 3. These loci, discovered by McMullen are GL^+(2,R)-invariant submanifolds (of complex dimension 3) that project to the locus of Riemann surfaces whose Jacobian variety has a factor admitting real multiplication by some quadratic order Ord_D. It turns out that these subvarieties can be classified by the discriminant D of the corresponding quadratic orders. However there algebraic varieties are not necessarily irreducible. …
The main goal of this work is to construct and study a reasonable compactification of the strata of the moduli space of Abelian differentials. This allows us to compute the Kodaira dimension of some strata of the moduli space of Abelian differentials. The main ingredients to study the compactifications of the strata are a version of the plumbing cylinder construction for differential forms and an extension of the parity of the connected components of the strata to the differentials on curves of compact type. We study in detail the compactifications of the hyperelliptic minimal strata and of th…
We show that for many strata of Abelian differentials in low genus the sum of Lyapunov exponents for the Teichmueller geodesic flow is the same for all Teichmueller curves in that stratum, hence equal to the sum of Lyapunov exponents for the whole stratum. This behavior is due to the disjointness property of Teichmueller curves with various geometrically defined divisors on moduli spaces of curves.
We compute the Z/\ell and \ell-adic monodromy of every irreducible component of the moduli space M_g^f of curves of genus and and p-rank f. In particular, we prove that the Z/\ell-monodromy of every component of M_g^f is the symplectic group Sp_{2g}(Z/\ell) if g>=3 and \ell is a prime distinct from p. We give applications to the generic behavior of automorphism groups, Jacobians, class groups, and zeta functions of curves of given genus and p-rank.
We study genus 2 curves over finite fields of small characteristic. The $p$-rank $f$ of a curve induces a stratification of the coarse moduli space $\mathcal{M}_2$ of genus 2 curves up to isomorphism. We are interested in the size of those strata for all $f \in \{0,1,2\}$. In characteristic 2 and 3, previous results show that the supersingular $f=0$ stratum has size $q$. We show that for $q=3^r$, over $\mathbb{F}_q$ the non-ordinary $f=1$ and ordinary $f=2$ strata are of size $q(q-1)$ and $q^2(q-1)$, respectively. We give results found from computer calculations which suggest that these formul…
In this note, we give a so-called representative classification for the strata by automorphism group of smooth $\bar{k}$-plane curves of genus $6$, where $\bar{k}$ is a fixed separable closure of a field $k$ of characteristic $p = 0$ or $p > 13$. We start with a classification already obtained by the first author and we use standard techniques. Interestingly, in the way to get these families for the different strata, we find two remarkable phenomenons that did not appear before. One is the existence of a non $0$-dimensional final stratum of plane curves. At a first sight it may sound odd, but …
Suppose $X$ is a hyperelliptic curve of genus $g$ defined over an algebraically closed field $k$ of characteristic $p=2$. We prove that the de Rham cohomology of $X$ decomposes into pieces indexed by the branch points of the hyperelliptic cover. This allows us to compute the isomorphism class of the $2$-torsion group scheme $J_X[2]$ of the Jacobian of $X$ in terms of the Ekedahl-Oort type. The interesting feature is that $J_X[2]$ depends only on some discrete invariants of $X$, namely, on the ramification invariants associated with the branch points. We give a complete classification of the gr…
Let k=F_q be a finite field of even characteristic. We obtain in this paper a complete classification, up to k-isomorphism, of non singular quartic plane curves defined over k. We find explicit rational normal models and we give closed formulas for the total number of k-isomorphism classes. We deduce from these computations the number of k-rational points of the different strata by the Newton polygon of the non hyperelliptic locus M_3^{nh} of the moduli space M_3 of curves of genus 3. By adding to these computations the knowed results on the hyperelliptic locus we obtain a complete picture of …
We prove that the Poincare' polynomial of the moduli space of smooth genus 4 curves is 1+t^2+t^4+t^5. We show this by producing a stratification of the space, such that all strata are geometric quotients of complements of discriminants.
We study the moduli space $\mathcal{AS}_{g}$ of Artin-Schreier curves of genus $g$ over an algebraically closed field $k$ of positive characteristic $p$. The moduli space is partitioned by irreducible strata, where each stratum parameterizes Artin-Schreier curves whose ramification divisors have the same coefficients. We construct deformations of these curves to study the relations between those strata. As an application, when $p=3$, we prove that $\mathcal{AS}_{g}$ is connected for all possible $g$. When $p>3$, it turns out that $\mathcal{AS}_{g}$ is connected for sufficiently large value of …
The discriminant, $D$, in the base of a miniversal deformation of an irreducible plane curve singularity, is partitioned according to the genus of the (singular) fibre, or, equivalently, by the sum of the delta invariants of the singular points of the fibre. The members of the partition are known as the {\it Severi strata}. The smallest is the $δ$-constant stratum, $D(δ)$, where the genus of the fibre is $0$. It is well known, by work of Givental' and Varchenko, to be Lagrangian with respect to the symplectic form $Ω$ obtained by pulling back the intersection form on the cohomology of the fibr…
Inside the moduli space of curves of genus three with one marked point, we consider the locus of hyperelliptic curves with a marked Weierstrass point, and the locus of non-hyperelliptic curves with a marked hyperflex. These loci have codimension two. We compute the classes of their closures in the moduli space of stable curves of genus three with one marked point. Similarly, we compute the class of the closure of the locus of curves of genus four with an even theta characteristic vanishing with order three at a certain point. These loci naturally arise in the study of minimal strata of Abelian…
Let L be a quadratic imaginary field, inert at the rational prime p. Fix an integer n at least 3, and let M be the moduli space (in characteristic p) of principally polarized abelian varieties of dimension n equipped with an action by O_L of signature of (1,n-1). We show that each Newton stratum of M, other than the supersingular stratum, is irreducible.
A new codimension 2 relation among descendent strata in the moduli space of stable, 3-pointed, genus 2 curves is found. The space of pointed admissible double covers is used in the calculation. The resulting differential equations satisfied by the genus 2 gravitational potentials of varieties in Gromov-Witten theory are described. These are analogous to the WDVV-equations in genus 0 and Getzler's equations in genus 1. As an application, genus 2 descendent invariants of the projective plane are determined, including the classical genus 2 Severi degrees.
In the 80's H. Masur and W. Veech defined two numerical invariants of strata of abelian differentials: the volume and the Siegel-Veech constant. Based on numerical experiments, A. Eskin and A. Zorich proposed a series of conjectures for the large genus asymptotics of these invariants. By a careful analysis of the asymptotic behavior of quasi-modular forms, D. Chen, M. Moeller, and D. Zagier proved this conjecture for strata of differentials with simple zeros. Here, we prove that the conjecture holds for the other extreme case, i.e. for strata of differentials with a unique zero. Our main ingre…
A $k$-differential on a Riemann surface is a section of the $k$-th power of the canonical bundle. Loci of $k$-differentials with prescribed number and multiplicities of zeros and poles form a natural stratification for the moduli space of $k$-differentials. The classification of connected components of the strata of $k$-differentials was known for holomorphic differentials, meromorphic differentials and quadratic differentials with at worst simple poles by Kontsevich--Zorich, Boissy and Lanneau, respectively. Built on their work we develop new techniques to study connected components of the st…
We introduce a new family of tautological relations of the moduli space of stable curves of genus $g$. These relations are obtained by computing the Poincaré-dual class of empty loci in the Hodge bundle. We use these relations to obtain a new expression for the Chern classes of the Hodge bundle. We prove that the $(g-i)$th class can be expressed as a linear combination of tautological classes involving only stable graphs with at most $i$ loops. In particular the top Chern class may be expressed with trees. This property was expected as a consequence of the DR/DZ equivalence conjecture by Burya…
Let M_{g,n} be the moduli space of stable genus g curves with n marked points. M_{g,n} has boundary strata consisting of nodal curves. The fundamental classes of these boundary strata may be linearly dependent in the Chow group A_*(M_{g,n}). Relations among these boundary strata can be found by exploiting a localization trick involving stable maps to P^1. This note explains this trick and applies it to give a new derivation of Getzler's relation among codimension 2 boundary strata in M_{1,4}.
Local properties of families of algebraic subsets $W_g$ in Birkhoff strata $Σ_{2g}$ of Gr$^{(2)}$ containing hyperelliptic curves of genus $g$ are studied. It is shown that the tangent spaces $T_g$ for $W_g$ are isomorphic to linear spaces of 2-coboundaries. Particular subsets in $W_g$ are described by the intergrable dispersionless coupled KdV systems of hydrodynamical type defining a special class of 2-cocycles and 2-coboundaries in $T_g$. It is demonstrated that the blows-ups of such 2-cocycles and 2-coboundaries and gradient catastrophes for associated integrable systems are interrelated.
We compute cohomology of the moduli space of genus three curves with level two structure and some related spaces. In particular, we determine the cohomology groups of the moduli space of plane quartics with level two structure as representations of the symplectic group on a six dimensional vector space over the field of two elements. We also make the analogous computations for some related spaces such as moduli spaces of genus three curves with a marked points and strata of the moduli space of Abelian differentials of genus three.
This paper lays the foundation for determining the Kodaira dimension of the projectivized strata of Abelian differentials with prescribed zero and pole orders in large genus. We work with the moduli space of multi-scale differentials constructed in [BCGGM2] which provides an orbifold compactification of these strata. We establish the projectivity of the moduli space of multi-scale differentials, describe the locus of canonical singularities, and compute a series of effective divisor classes. Moreover, we exhibit a perturbation of the canonical class which allows the corresponding pluri-canonic…
Modular curves $X_{1}(N)$ parametrize elliptic curves with a point of order $N$. They can be identified with connected components of projectivized strata $\mathbb{P}\mathcal{H}(a,-a)$ of meromorphic differentials. As strata of meromorphic differentials, they have a canonical walls-and-chambers structure defined by the topological changes in the flat structure defined by the meromorphic differentials. We provide formulas for the number of chambers and an effective means for drawing the incidence graph of the chamber structure of any modular curve $X_{1}(N)$. This defines a family of graphs with…
We introduce the moduli space $R \bar{M}_{2k,l}$ of pointed real curves of genus zero and give its natural stratification. The strata of $R \bar{M}_{2k,l}$ correspond to real curves of genus zero with different degeneration types and are encoded by their dual trees with certain decorations. By using this stratification, we calculate the first Stiefel-Whitney class of $R \bar{M}_{2k,l}$ and construct the orientation double cover $R \widetilde{M}_{2k,l}$$ of $R \bar{M}_{2k,l}$.
We show that the locally closed strata of the Weierstrass flags on the moduli spaces of curves of genus g and on the moduli space of curves of genus g with one marked point are almost never affine.
Let $\mathcal{A}_{g}$ be the moduli space of $g$-dimensional principally polarized abelian varieties over $\mathbb{Z}$, and let $\mathcal{T} \subset \mathcal{A}_{g}$ be a closed locus, also defined over $\mathbb{Z}$. Motivated by unlikely intersection conjectures, we study the intersection of $\mathcal{T}_{\mathbb{F}_{p}}$ with the Bruhat strata in $\mathcal{A}_{g,\mathbb{F}_{p}}$ as $p$-varies; these are strata characterized by the existence of certain subgroup schemes inside the $p$-torsion of the fibres. We find that, away from a finite set of primes, positive-dimensional ``unlikely'' inter…
Let $\mathcal{M}_{g}$ be the moduli space of compact connected hyperbolic surfaces of genus $g\geq2$, and ${\mathcal B}_g \subset {\mathcal M}_{g} $ its branch locus. Let $\widehat{\mathcal{M}_{g}}$ be the Deligne-Mumford compactification of the moduli space of smooth, complete, connected surfaces of genus $g\geq 2$ over $\mathbb{C}$. The branch locus ${\mathcal B}_g$ is stratified by smooth locally closed equisymmetric strata, where a stratum consists of hyperbolic surfaces with equivalent action of their preserving orientation isometry group. Any stratum can be determined by a certain epimor…
The moduli space $\bar{M}_A$ of weighted pointed stable curves of genus zero is stratified according to the degeneration types of such curves. We show that the homology groups of the moduli space $\bar{M}_A$ are generated by the strata of $\bar{M}_A$ and give all additive relations between them. We also observe that the Chow groups $A_i \bar{M}_A$ and the homology groups $H_{2i} \bar{M}_A$ are isomorphic. This generalizes Kontsevich-Manin's and Losev-Manin's theorems to arbitrary weight data $A$.
We obtain a simple, recursive presentation of the tautological (κ, ψ, and λ) classes on the moduli space of curves in genus zero and one in terms of boundary strata (graphs). We derive differential equations for the generating functions for their intersection numbers which allow us to prove a simple relationship between the genus zero and genus one potentials. As an application, we describe the moduli space of normalized, even, rank one cohomological field theories in genus one in coordinates which are additive under taking tensor products. Our results simplify and generalize those of Kaufmann…
