GL(2,R)-Invariant Measures in Marked Strata: Generic Marked Points, Earle-Kra for Strata, and Illumination

TL;DR

This paper classifies GL(2,R) invariant point markings in strata of Abelian differentials, revealing they only occur in hyperelliptic components and relate to Weierstrass points, with applications to holomorphic sections and the finite blocking problem.

Contribution

It provides a complete classification of invariant point markings, linking them to hyperelliptic structures and demonstrating their use in understanding holomorphic sections and translation surface dynamics.

Findings

Invariant point markings only exist in hyperelliptic components.

Markings correspond to Weierstrass points or pairs exchanged by involution.

Finite blocking problem is solved for dense GL(2,R) orbit surfaces.

Abstract

We classify GL(2,R) invariant point markings over components of strata of Abelian differentials. Such point markings exist only when the component is hyperelliptic and arise from marking Weierstrass points or two points exchanged by the hyperelliptic involution. We show that these point markings can be used to determine the holomorphic sections of the universal curve restricted to orbifold covers of subvarieties of the moduli space of Riemann surfaces that contain a Teichmuller disk. The finite blocking problem is also solved for translation surfaces with dense GL(2,R) orbit.