Stability of bicanonical curves of genus three

Abstract

We work out the geometric invariant theory of the Hilbert scheme and the Chow variety of bicanonically embedded curves of genus three. We show that a Hilbert semistable bicanonical curve has node, ordinary cusp and tacnode as Singularity but does not admit elliptic tails or bridges, and that it is Hilbert stable if, moreover, it does not have a tacnode. We prove that a Chow semistable bicanonical curve allows the same set of singularities but no elliptic tail, and that those with an elliptic bridge or a tacnode are identified ill the GIT quotient space. We also give a geometric description of the birational morphism from the Hilbert quotient to the Chow quotient. (C) 2009 Elsevier B.V. All rights reserved.