${\pi }_1$ of trigonal loci of strata of abelian differentials

Abstract

We investigate locally closed subspaces of projectivized strata of abelian differentials which classify trigonal curves with canonical divisor a multiple of a trigonal divisor. We describe their orbifold structure using linear systems on Segre-Hirzebruch surfaces and obtain results for their orbifold fundamental groups. Most notable among these orbifolds is the connected component $\mathbf{P}{\mathcal{H}}_4^{ev}(6)$, the projectivisation of the space ${\mathcal{H}}_4^{ev}(6)$ of abelian differentials on non-hyperelliptic genus $4$ curves with a single zero of multiplicity 6 providing an even spin structure. Its orbifold fundamental group is identified with the quotient of the Artin group of type $E_8$ by its maximal central subgroup.

arXiv comment13 pages, in v2 a reference was added to acknowledge a previous proof of a claim in the preprint