Strata of abelian differentials and the Teichmüller dynamics

Curves of genus $g$ which admit a map to $\mathbf {P}^{1}$ with specified ramification profile $\mu$ over $0\in \mathbf {P}^{1}$ and $\nu$ over $\infty\in \mathbf {P}^{1}$ define a double ramification cycle $\mathsf{DR}_{g}(\mu,\nu)$ on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramification cycles on the moduli space of Deligne-Mumford stable curves. The cycle $\mathsf{DR}_{g}(\mu,\nu)$ for stable curves is defined via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for $\mathsf{DR}_{g}(\mu,\nu)$ in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramification cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hain’s formula in the compact type case. When $\mu=\nu=\emptyset$ , the formula for double ramification cycles expresses the top Chern class $\lambda_{g}$ of the Hodge bundle of $\overline {\mathcal{M}}_{g}$ as a push-forward of tautological classes supported on the divisor of non-separating nodes. Applications to Hodge integral calculations are given.

The purpose of this thesis is to study tautological rings of moduli spaces of curves. The moduli spaces of curves play an important role in algebraic geometry. The study of algebraic cycles on these spaces was started by Mumford. He introduced the notion of tautological classes on moduli spaces of curves. Faber and Pandharipande have proposed several deep conjectures about the structure of the tautological algebras. According to the Gorenstein conjectures these algebras satisfy a form of Poincare duality. The thesis contains three papers. In paper I we compute the tautological ring of the moduli space of stable n-pointed curves of genus one of compact type. We prove that it is a Gorenstein algebra. In paper II we consider the classical case of genus zero and its Chow ring. This ring was originally studied by Keel. He gives an inductive algorithm to compute the Chow ring of the space. Our new construction of the moduli space leads to a simpler presentation of the intersection ring and an explicit form of Keel’s inductive result. In paper III we study the tautological ring of the moduli space of stable n-pointed curves of genus two with rational tails. The Gorenstein conjecture is proved in this case as well.

We show that for many strata of Abelian differentials in low genus the sum of Lyapunov exponents for the Teichmuller geodesic flow is the same for all Teichmuller 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 Teichmuller curves with various geometrically defined divisors on moduli spaces of curves. 14H10; 37D40, 14H51

Square-tiled surfaces and rigid curves on moduli spaces

We compute many new classes of effective divisors in M¯g,n coming from the strata of Abelian differentials. Our method utilizes maps between moduli spaces and the degeneration of Abelian differentials.

We study contractions of the moduli space of stable curves beyond the minimal model ofM¯g′\overline {\mathcal {M}}_{g’}by giving a complete enumerative description of the rational map between two moduli spaces of curvesM¯g⇢M¯g′\overline {\mathcal {M}}_g \dashrightarrow \overline {\mathcal {M}}_{g’}which associates to a curveCCof genusggits Brill–Noether locus of special divisors in the case this locus is a curve. As an application we construct many examples of moving effective divisors onM¯g\overline {\mathcal {M}}_gof small slope, which in turn can be used to show that various moduli space of curves with level structure are of general type. For lowg′g’our calculation can be used to study the intersection theory of the moduli space of Prym varieties of dimension55.

Towards the geometry of double Hurwitz numbers

We define and study a natural system of tautological rings on the moduli spaces of marked curves at the level of differential forms. We show that certain 2-forms obtained from the natural normal functions on these moduli spaces are tautological. Also we show that rings of tautological forms are always finite dimensional. Finally we characterize the Kawazumi–Zhang invariant as essentially the only smooth function on the moduli space of curves whose Levi form is a tautological form.

In this article we provide a stack-theoretic framework to study the universal tropical Jacobian over the moduli space of tropical curves. We develop two approaches to the process of tropicalization of the universal compactified Jacobian over the moduli space of curves -- one from a logarithmic and the other from a non-Archimedean analytic point of view. The central result from both points of view is that the tropicalization of the universal compactified Jacobian is the universal tropical Jacobian and that the tropicalization maps in each of the two contexts are compatible with the tautological morphisms. In a sequel we will use the techniques developed here to provide explicit polyhedral models for the logarithmic Picard variety.Comment: 51 pages, 2 figures, v3: published version

We define a moduli space of rational curves with finite-order automorphism and weighted orbits, and we prove that the combinatorics of its boundary strata are encoded by a particular polytopal complex that also captures the algebraic structure of a complex reflection group acting on the moduli space. This generalizes the situation for Losev-Manin's moduli space of curves (whose boundary strata are encoded by the permutohedron and related to the symmetric group) as well as the situation for Batyrev-Blume's moduli space of curves with involution, and it extends that work beyond the toric context.

We study the transcendence of periods of abelian differentials, both at the arithmetic and functional level, from the point of view of the natural bi-algebraic structure on strata of abelian differentials. We characterize geometrically the arithmetic points, study their distribution, and prove that in many cases the bi-algebraic curves are the linear ones.

An abelian differential on a surface defines a flat metric and a vector field on the complement of a finite set of points. The vertical flow that can be defined on the surface has two kinds of invariant closed sets (i.e. invariant components) — periodic components and minimal components. We give upper bounds on the number of minimal components, on the number of periodic components and on the total number of invariant components in every stratum of abelian differentials. We also show that these bounds are tight in every stratum.

In this paper we describe the space spanned by the divisors of curves and points satisfying certain ramification conditions on ℳ¯ g,n , the moduli space of n-pointed stable curves of genus g. This generalizes work of Eisenbud, Harris, and Mumford for the cases n ≤ 1.

As pointed out in Arbarello and Cornalba (i>J. Alg. Geom. 5 (1996), 705–749), a theorem due to Di Francesco, Itzykson, and Zuber (see Di Francesco, Itzykson, and Zuber, i>Commun. Math. Phys. 151 (1993), 193–219) should yield new relations among cohomology classes of the moduli space of pointed curves. The coefficients appearing in these new relations can be determined by the algorithm we introduce in this paper.

This article discusses the connection between the matrix models and algebraic geometry. In particular, it considers three specific applications of matrix models to algebraic geometry, namely: the Kontsevich matrix model that describes intersection indices on moduli spaces of curves with marked points; the Hermitian matrix model free energy at the leading expansion order as the prepotential of the Seiberg-Witten-Whitham-Krichever hierarchy; and the other orders of free energy and resolvent expansions as symplectic invariants and possibly amplitudes of open/closed strings. The article first describes the moduli space of algebraic curves and its parameterization via the Jenkins-Strebel differentials before analysing the relation between the so-called formal matrix models (solutions of the loop equation) and algebraic hierarchies of Dijkgraaf-Witten-Whitham-Krichever type. It also presents the WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equations, along with higher expansion terms and symplectic invariants.