Loci of curves with prescribed automorphism group

                   Mark Watkins
                University of Bristol

In 2002, Magaard, Shaska, Shpectorov, and V\"olklein completely
classified the loci of genus 3 curves with a prescribed
automorphism group, and for genera up to 10 classified the loci
corresponding to large groups. Equations are generally available
only in genus 3.  We shall discuss their work, and give some
examples and computations with Magma that are related to this.
The classification method starts with some classical theory of
Riemann surfaces, and eventually reduces to a group-theoretic
computation. One of the main difficulties is to eliminate groups
which are a subgroup of a full automorphism group of a curve of a
given genus, but are themselves not the full automorphism group
of any such curve.