The locus of curves with prescribed automorphism group., by K. Magaard, T. Shaska, S. Shpectorov, and H. Voelklein.
Abstract: Let $G$ be a finite group, and $g\ge2$.
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 $g\le10$ we classify
those loci corresponding to ``large'' $G$.
K. Magaard, T. Shaska, S. Shpectorov, and H. Voelklein. <email@example.com>