$\pi_1$ of trigonal loci of strata of abelian differentials

Michael Lönne

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arXiv · 2510.06364

$\pi_1$ of trigonal loci of strata of abelian differentials

Michael Lönne

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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 $\mathbf P\mathcal H^{ev}_4(6)$, the projectivisation of the space $\mathcal H^{ev}_4(6)$ 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.

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Michael Lönne. 2025-10-07. $\pi_1$ of trigonal loci of strata of abelian differentials. https://arxiv.org/abs/2510.06364

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