Effective Divisors in M¯g,n from Abelian Differentials

This paper focuses on the interplay between the intersection theoryand the Teichmüller dynamics on the moduli space of curves. Asapplications, we study the cycle class of strata of the Hodge bundle,present an algebraic method to calculate the class of the divisorparameterizing abelian differentials with a nonsimple zero, andverify a number of extremal effective divisors on the moduli space ofpointed curves in low genus.

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.

On the effective cone of [formula omitted

We show that the Masur–Veech volumes and area Siegel–Veech constants can be obtained using intersection theory on strata of Abelian differentials with prescribed orders of zeros. As applications, we evaluate their large genus limits and compute the saddle connection Siegel–Veech constants for all strata. We also show that the same results hold for the spin and hyperelliptic components of the strata.

The Neron-Severi group of divisor classes modulo algebraic equivalence on a smooth algebraic surface is often not difficult to calculate, and has classically been studied as one of the fundamental invariants of the surface. A more difficult problem is the determination of those divisor classes which can be represented by effective divisors; these divisor classes form a monoid contained in the Neron-Severi group. Despite the finite generation of the whole Neron-Severi group, the monoid of effective divisor classes may or may not be finitely generated, and the methods used to explicitly calculate this monoid seem to vary widely as one proceeds from one type of surface to another in the standard classification scheme (see Rosoff, 1980, 1981). In this paper we shall use concrete vector bundle techniques to describe the monoid of effective divisor classes modulo algebraic equivalence on a complex ruled surface over a given base curve. We will find that, over a base curve of genus 0, the monoid of effective divisor classes is very simple, having two generators (which is perhaps to be expected), while for a ruled surface over a curve of genus 1, the monoid is more complicated, having either two or three generators. Over a base curve of genus 2 or greater, we will give necessary and sufficient conditions for a ruled surface to have its monoid of effective divisor classes finitely generated; these conditions point to the existence of many ruled surfaces over curves of higher genus for which finite generation fails.

Cette note donne une preuve élémentaire que les strates des différentiels abéliens ne contiennent pas de variétés algébriques complètes.

On the space of ergodic measures for the horocycle flow on strata of Abelian differentials

We show that an algebraic subvariety of the moduli space of genus g Riemann surfaces is coarsely dense with respect to the Teichmüller metric (or Thurston metric) if and only if it has full dimension. We apply this to determine which strata of abelian differentials have coarsely dense projection to moduli space. Furthermore, we prove a result on coarse density of projections of GL 2 (ℝ)-orbit closures in the space of abelian differentials.

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

A cyclic cover of the complex projective line branched at four appropriate points has a natural structure of a square-tiled surface. We describe the combinatorics of such a square-tiled surface, the geometry of the corresponding Teichmüller curve, and compute the Lyapunov exponents of the determinant bundle over the Teichmüller curve with respect to the geodesic flow. This paper includes a new example (announced by G. Forni and C. Matheus in [17] of a Teichmüller curve of a square-tiled cyclic cover in a stratum of Abelian differentials in genus four with a maximally degenerate Kontsevich--Zorich spectrum (the only known example in genus three found previously by Forni also corresponds to a square-tiled cyclic cover [15]. We present several new examples of Teichmüller curves in strata of holomorphic and meromorphic quadratic differentials with a maximally degenerate Kontsevich--Zorich spectrum. Presumably, these examples cover all possible Teichmüller curves with maximally degenerate spectra. We prove that this is indeed the case within the class of square-tiled cyclic covers.

We introduce a twisted cohomology cocycle over the Teichmüller flow and prove a “spectral gap” for its Lyapunov spectrum with respect to the Masur–Veech measures. We then derive Hölder estimates on spectral measures and bounds on the speed of weak mixing for almost all translation flows in every stratum of Abelian differentials on Riemann surfaces, as well as bounds on the deviation of ergodic averages for product translation flows on the product of a translation surface with a circle.

Abelian differentials on Riemann surfaces can be seen as translation surfaces, which are flat surfaces with cone-type singularities. Closed geodesics for the associated flat metrics form cylinders whose number under a given maximal length generically has quadratic asymptotics in this length, with a common coefficient constant for the quadratic asymptotics called a Siegel--Veech constant which is shared by almost all surfaces in each moduli space of translation surfaces. Square-tiled surfaces are specific translation surfaces which have their own quadratic asymptotics for the number of cylinders of closed geodesics. It is an interesting question whether, as n tends to infinity, the Siegel--Veech constants of square-tiled surfaces with n tiles tend to the generic constants of the ambient moduli space. We prove that this is the case in the moduli space H(2) of translation surfaces of genus two with one singularity.

Let $X$ be a smooth projective surface defined over the complex number field and let $D$ be an effective divisor on $X$. In this paper we will propose a special class of effective divisors which has some properties similar to that of the case where $D$ is ample and we will study this divisor.

This thesis investigates various questions concerning the birational geometry of the moduli spaces Mg and Mg,n, with a focus on the computation of effective divisor classes. In Chapter 2 we define, for any n-tuple d of integers summing up to g − 1, a geometrically meaningful divisor on Mg,n that is essentially the pullback of the theta divisor on a universal Jacobian variety under an Abel-Jacobi map. It is a generalization of various kinds of divisors used in the literature, for example by Logan to show that Mg,n is of general type for all g ≥ 4 as soon as n is big enough. We compute the class of this divisor and show that for certain choices of d it is irreducible and extremal in the effective cone of Mg,n. Chapter 3 deals with a birational model X6 of M6 that is obtained by taking quadric hyperplane sections of the degree 5 del Pezzo surface. We compute the class of the big divisor inducing the birational map M6 99K X6 and use it to derive an upper bound on the moving slope of M6. Furthermore we show that X6 is the final non-trivial space in the log minimal model program for M6. We also give a few results on the unirationality of Weierstras loci on Mg,1, which for g = 6 are related to the del Pezzo construction used to construct the model X6. Finally, Chapter 4 focuses on the case g = 0. Castravet and Tevelev introduced combinatorially defined hypertree divisors on M0,n that for n = 6 generate the effective cone together with boundary divisors. We compute the class of the hypertree divisor on M0,7, which is unique up to permutation of the marked points. We also give a geometric characterization of it that is analogous to the one given by Keel and Vermeire in the n = 6 case.