CANONICAL CURVES IN ℙ³

Abstract

Let ${\rm Hilb}^{6t-3}(\mathbb{P}^3)$ be the Hilbert scheme of closed 1-dimensional subschemes of degree 6 and arithmetic genus 4 in $\mathbb{P}^3$. Let $H$ be the component of ${\rm Hilb}^{6t-3}(\mathbb{P}^3)$ whose generic point corresponds to a canonical curve, that is, a complete intersection of a quadric and a cubic surface in $\mathbb{P}^3$. Let $F$ be the vector space of linear forms in the variables $z_1, z_2, z_3, z_4$. Denote by $F_d$ the vector space of homogeneous forms of degree $d$. Set $X = \{(f_2,f_3)\}$ where $f_2 \in \mathbb{P}(F_2)$ is a quadric surface, and $f_3 \in \mathbb{P}(F_3/f_2 \cdot F)$ is a cubic modulo $f_2$. We have a rational map, $\sigma : X \cdots \rightarrow H$ defined by $(f_2,f_3) \mapsto f_2 \cap f_3$. It fails to be regular along the locus where $f_2$ and $f_3$ acquire a common linear component. Our main result gives an explicit resolution of the indeterminacies of $\sigma$ as well as of the singularities of $H$. 2000 Mathematical Subject Classification: 14C05, 14N05, 14N10, 14N15.

Log canonical models and variation of GIT for genus 4 canonical curves

“…The Rojas-Vainsencher resolution. Rojas-Vainsencher [RV02] have constructed an explicit resolution W of the rational map PE Hilb 4,1 , giving a diagram:…”

The geometry of the ball quotient model of the moduli space of genus four curves

“…Such curves are the complete intersection of a quadric and cubic in P 3 . Although these complete intersections can be parameterized naturally by a subset of the Hilbert scheme, or Chow variety, we find it is more convenient to work with the closely related projective bundle PE parameterizing subschemes of P 3 with ideal defined by a quadric and cubic ( §1.1; for the relation to the Hilbert scheme see [RV02]). The GIT quotient we consider is induced by a GIT problem for cubic threefolds.…”