CHOW STABILITY OF CANONICAL GENUS 4 CURVES

Abstract

In this paper, we give sufficient conditions on a canonical genus 4 curve for it to be Chow (semi)stable. A Deligne-Mumford stable curve is a complete connected curve C having ample dualising sheaf !C and admitting only nodes as singularities. An n- canonical curve C ⊂ P N is a Deligne-Mumford stable curve of arithmetic genus g embedded by the complete linear system |! ⊗n C | where N = (2n−1)(g−1)−1 if n ≥ 2, and N = g − 1 if n = 1. Let Chowg,n be the closure of the locus of the Chow forms of n-canonical curves of arithmetic genus g in the Chow variety of algebraic cycles of dimension 1 and degree 2g−2 in P N. The natural action of SLN+1 on P N induces an action on Chowg,n. Denote the corresponding GIT (Geometric Invariant Theory) quotient space by Chowg,n//SLN+1. To understand this quotient space as a parameter space of curves with some geometric properties, we need to find Chow stability conditions. Mumford showed that, for n ≥ 5 and g ≥ 2, the Chow stable curves are pre- cisely Deligne-Mumford stable curves and there is no strictly Chow semistable curve (cf. (14)). This implies that the quotient space is precisely the moduli space of Deligne-Mumford stable curves M4. The cases when n = 3 and g ≥ 3 were concerned by Schubert in (16). He proved that a 3-canonical curve of genus g ≥ 3 is Chow stable if and only if it is pseudo-stable and also showed that there is no strictly Chow semistable curve, and thus the quotient space is the moduli space of pseudo-stable curves M ps . A pseudo-stable curve is a complete connected curve C satisfying the following properties.