Stability of 2nd Hilbert Points of Canonical Curves

Abstract

We establish GIT semistability of the 2 nd Hilbert point of every Gieseker-Petri general canonical curve by a simple geometric argument. As a consequence, we obtain an upper bound on slopes of general families of Gorenstein curves. We also explore the question of what replaces hyperelliptic curves in the GIT quotients of the Hilbert scheme of canonical curves.

The Final Log Canonical Model of the Moduli Space of Stable Curves of Genus 4

“…We proceed to describe geometrically meaningful strata inside the stable locus: 2.3.7. Curves with A 8 and A 9 singularities: Consider an A 8 -curve C (see [FJ11] for a general background on canonical A-curves) defined parametrically by…”

On the Kodaira dimension of Hurwitz spaces

“…It is furthermore well known that I 2 (ω C ) = I 2 (O R (1)), that is, the second Hilbert point of the canonical curve C and that of the scroll R coincide. Furthermore, I 2 (O R (1)) is semistable, see [FJ,Proposition 3.1]. We now choose a divisor We now establish part (ii) of our claim.…”

“…g (1) the line bundle on the GIT quotient induced by the Pl ücker line bundle on G g−2 2 , Sym 2 C g . The fact that the second Hilbert point of a general curve C is semistable and thus ϕ 2 is well defined is established in [FJ,Theorem 1.1].…”

Stability Conditions and Positivity of Invariants of Fibrations

“…The Cornalba-Harris method can be applied with this kind of assumption. For instance we can prove the following result (cf [22] for h = 2). Suppose that the h-th Hilbert point of a general fibre F with its canonical sheaf is semistable (with h ≥ 2).…”