Strata Of Quadratic Differentials Research Articles

Marked points on translation surfaces

We show that all GL(2,R) equivariant point markings over orbit closures of translation surfaces arise from branched covering constructions and periodic points, completely classify such point markings over strata of quadratic differentials, and give applications to the finite blocking problem.

Lyapunov exponents of the Hodge bundle over strata of quadratic differentials with large number of poles

We show an upper bound for the sum of positive Lyapunov exponents of any Teichmuller curve in strata of quadratic differentials with at least one zero of large multiplicity. As a corollary, it holds for any $SL(2,\mathbb R)$-invariant subspaces defined over $\mathbb Q$ in these strata. This proves Grivaux-Hubert's conjecture about the asymptotics of Lyapunov exponents for strata with a large number of poles in the situation when at least one zero has large multiplicity.

GL+(2, ℝ)–orbits in Prym eigenform loci

This paper is devoted to the classification of GL^+(2,R)-orbit closures of surfaces in the intersection of the Prym eigenform locus with various strata of quadratic differentials. We show that the following dichotomy holds: an orbit is either closed or dense in a connected component of the Prym eigenform locus. The proof uses several topological properties of Prym eigenforms, which are proved by the authors in a previous work. In particular the tools and the proof are independent of the recent results of Eskin-Mirzakhani-Mohammadi. As an application we obtain a finiteness result for the number of closed GL^+(2,R)-orbits (not necessarily primitive) in the Prym eigenform locus Prym_D(2,2) for any fixed D that is not a square.

Quadratic differentials in low genus: exceptional and non-varying strata

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 Teichmuller 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 Teichmuller geodesic flow is the same for all Teichmuller curves in that stratum.

Degenerations of quadratic differentials on ℂℙ1

We describe the connected components of the complement of a natural "diagonal" of real codimension 1 in a stratum of quadratic differentials on CP1. We establish a natural bijection between the set of these connected components and the set of generic configurations that appear on such "flat spheres". We also prove that the stratum has only one topological end. Finally, we elaborate a necessary toolkit destined to evaluation of the Siegel-Veech constants.