Connected components of strata of quadratic differentials over Teichmuller space

TL;DR

This paper investigates the connected components of strata of quadratic differentials over Teichmuller space, establishing bounds and exact counts using line bundle sections and a generalized Gauss map invariant.

Contribution

It introduces bounds and exact counts for the number of connected components of quadratic differential strata using novel geometric invariants.

Findings

Upper bounds on the number of components

Lower bounds via generalized Gauss map

Exact counts for strata with many zeros of the same order

Abstract

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 the number of such components. For strata with sufficiently many zeroes of the same order we can state precisely the number of components.