arXiv research

The paper proves properties of strata of differentials, showing they are affine and extremal.

For $g\geq2$, $j=1,\dots,g$ and $n\geq g+j$ we exhibit infinitely many new rigid and extremal effective codimension $j$ cycles in $\overline{\mathcal{M}}_{g,n}$ from the strata of quadratic differentials and projections of these strata under forgetful morphisms and show the same holds for $k$-differentials with $k\geq …

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 asymp…

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-simp…

For every $g\geq 2$ and $n\geq g+1$ we exhibit infinitely many extremal effective divisors in $\overline{\mathcal{M}}_{g,n}$ coming from the strata of abelian differentials.

Study shows non-polyhedral structure in moduli spaces for n≥8.

We compute the class of the closure of the locus of canonical divisors in the projectivization of the Hodge bundle $\mathbb{P}\overline{\mathcal{H}}_g$ over $\overline{\mathcal{M}}_g$ which have a zero at a Weierstrass point. We also show that the strata of canonical and bicanonical divisors with a double zero span ext…

This paper classifies components of meromorphic differential strata.

Classifies connected components of meromorphic differentials with residue conditions.

Study quantifies convergence of Alexandrov spaces without collapsing.

Classifies orbit closures in translation surface strata.

Formula for volumes of odd strata of quadratic differentials using graph intersection numbers.

Harmonic maps to Euclidean buildings have rectifiable singular strata.

Study classifies Morse functions with 4 critical points on immersed 2-spheres.

Constructs a function to prove meromorphic differential strata don't have complete subvarieties.

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 ar…

The paper constructs cohomology classes on curve strata.

The paper studies tautological rings of strata of differentials, proving cohomological stability and no relations in specific degrees.

Alternative proof for non-existence of complete curves in differential strata.

Homology of abelian differentials stabilizes with more zeros.

Survey of compactifications for differential strata.

We prove a number of convexity results for strata of the diagonal pants graph of a surface, in analogy with the extrinsic geometric properties of strata in the Weil-Petersson completion. As a consequence, we exhibit convex flat subgraphs of every possible rank inside the diagonal pants graph.

Study linear subvarieties of meromorphic differential strata, proving toric closures and new proofs of theorems.

Non-asphericity of strata of genus-one differentials

Classifies Teichmüller curves in specific hyperelliptic components of meromorphic differentials.

We consider the nilpotent left-invariant sub-Riemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point (in particular, of the kinematic models of a car with a trailer). On the other hand, this is the …

In this paper, we study fundamental groups of strata of the moduli space of quadratic differentials. We use certain properties of the Abel-Jacobi map, combined with local surgeries on quadratic differentials, to construct quotient groups of the fundamental groups for a particular family of strata.

Convex iso-Delaunay regions found in flat surface strata.

The paper describes a cover of strata of k-differentials with a formula for fiber cardinality.

We investigate specific examples of locally-defined real vector-fields on strata of translation surfaces. Integrating SL(2,R)-loci of Veech surfaces along these vector-fields yield interesting new examples of horocyle-invariant ergodic measures. These measures are supported on closed immersed manifolds with boundary th…

Classifies components of strata of k-differentials on Riemann surfaces.

We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include the cla…

SageMath package diffstrata calculates intersection theory on abelian differentials.

Study uses graph techniques to understand meromorphic quadratic differential strata.

Study describes splitting and filtration of Hodge bundle on quadratic differentials.

The volumes of strata of Abelian or quadratic differentials play an important role in the study of dynamics on flat surfaces, related to dynamics in polygonal billiards. This article reviews all known ways to compute volumes in the quadratic case and provides explicit values of volumes of the strata of meromorphic quad…

Study counts ergodic measures in surface lamination strata.

We show that the Masur-Veech volumes and area Siegel-Veech constants can be obtained by intersection numbers on the 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…

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 bund…

The study characterizes a complex curve of residueless meromorphic differentials on elliptic curves.

An Abelian differential gives rise to a flat structure (translation surface) on the underlying Riemann surface. In some directions the directional flow on the flat surface may contain a periodic region that is made up of maximal cylinders filled by parallel geodesics of the same length. The growth rate of the number of…

New measure defined on surface strata, invariant under scaling.

Study of polynomial strata using braid groups and translation surfaces.

In this paper, we study connected components of strata of the space of quadratic differentials lying over $\T_g$. We use certain general properties of sections of line bundles to put a upper bound on the number of connected components, and a generalized version of the Gauss map as an invariant to put a lower bound on t…

Paper calculates distances between strata in Teichmüller space, proving a constant separation.

Formula for Masur-Veech volumes in quadratic differentials with odd zeros.

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 $η$…

Very few results are known about the topology of the strata of the moduli space of quadratic differentials. In this paper, we prove that any connected component of such strata has only one topological end. A typical flat surface in a neighborhood of the boundary is naturally split by a collection of parallel short sadd…