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 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 of the m th Hilbert points of curves X with elliptic tails. We compute the exact weight with which 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 ps g . Our computations yield, as byproducts, examples of both m-Hilbert unstable and m-Hilbert stable X that are Chow strictly semistable.