STABILITY OF TAILS AND 4-CANONICAL MODELS

Abstract

We show that the GIT quotients of suitable loci in the Hilbert and Chow schemes of 4-canonically embedded curves of genus g >= 3 are the moduli space (M) over bar (ps)(g) of pseudostable curves constructed by Schubert in [10] using Chow varieties and 3-canonical models. The only new ingredient needed in the Hilbert scheme variant is a more careful analysis of the stability with respect to a certain 1-ps lambda of the m(th) Hilbert points of curves X with elliptic tails. We compute the exact weight with which lambda acts, and not just the leading term in m of this weight. A similar analysis of stability of curves with rational cuspidal tails allows us to determine the stable and semistable 4-canonical Chow loci. Although here the geometry of the quotient is more complicated because there are strictly semistable orbits, we are able to again identify it as (M) over bar (ps)(g). Our computations yield, as byproducts, examples of both m-Hilbert unstable and m-Hilbert stable X that are Chow strictly semistable.