The locus of curves with prescribed automorphism group
K. Magaard, T. Shaska, S. Shpectorov, H. Voelklein
TL;DR
The paper develops an algebraic-geometry framework to study curves of genus g with prescribed automorphism groups by analyzing Hurwitz spaces and their images in moduli space, using dimension arguments to identify when a subgroup action extends to a full group action. It provides a practical criterion (Theorem 2) to determine full automorphism groups and applies it to yield a complete genus-3 classification with explicit curve equations, as well as a catalog of large automorphism groups up to genus 10. The approach combines group-theoretic generation data with moduli-theoretic constraints, and leverages computational tools (GAP, BRAID) to compute braid orbits and locus structures. The results advance understanding of how symmetries constrain the geometry of curves, with explicit models and locus dimensions that facilitate further study of automorphism-defining fields and moduli definitions. Formally, for a group G acting on genus g curves, the maximal automorphism-locus dimension is governed by the signature via δ(g,G, C) = 3g0 − 3 + r, and large groups necessarily yield orbit genus g0 = 0, enabling concrete equations and tabled classifications up to genus 10.
Abstract
Let G be a finite group, and $g \geq 2$. We study the locus of genus g curves that admit a G-action of given type, and inclusions between such loci. We use this to study the locus of genus g curves with prescribed automorphism group G. We completely classify these loci for g=3 (including equations for the corresponding curves), and for $g \leq 10$ we classify those loci corresponding to "large" G.