Some special families of hyperelliptic curves

Abstract

Let ${Ł}_g^G$ denote the locus of hyperelliptic curves of genus $g$ whose automorphism group contains a subgroup isomorphic to $G$. We study spaces ${Ł}_g^G$ for $G \iso \Z_n, \Z_2ø\Z_n, \Z_2øA_4$, or $S{L}_2(3)$. We show that for $G \iso \Z_n, \Z_2ø\Z_n$, the space ${Ł}_g^G$ is a rational variety and find generators of its function field. For $G\iso \Z_2øA_4, SL_2(3)$ we find a necessary condition in terms of the coefficients, whether or not the curve belongs to ${Ł}_g^G$. Further, we describe algebraically the loci of such curves for $g\le 12$ and show that for all curves in these loci the field of moduli is a field of definition.

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