$π_1$ of trigonal loci of strata of abelian differentials
Michael Lönne
TL;DR
The paper studies trigonal loci inside projectivized strata ${\mathbf P}\mathcal{H}_g$ of abelian differentials, focusing on strictly trigonal pairs $(C,D)$ where $D$ is an integral multiple of a trigonal divisor. It models these loci as discriminant complements in linear systems on Segre–Hirzebruch surfaces, achieving explicit global-quotient descriptions for the three components of ${\mathbf P}\mathcal{H}^{tri}_{3k+1}$ and relating their orbifold fundamental groups to discriminant knot groups; in genus $4$ the even-spin component has orbifold fundamental group $\pi_1^{orb} \cong Ar(E_8)/Centre$. The approach combines canonical geometry of trigonal curves with a sequence of coordinate transformations to reduce to linear-slice problems, enabling concrete topological computations and connections to braid-group theories via secondary braid groups. These results illuminate the link between the geometry of abelian-differential moduli, discriminant theory, and the topology of associated braid groups, and they propose a conjectural secondary-braid description for the remaining component. Overall, the work provides explicit orbifold-quotient models and topological invariants for trigonal loci, with special emphasis on the genus $4$ case and the interplay with $E_8$-type structures.
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.