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. <tshaska@math.uci.edu>