Compactifying the space of elliptic quartic curves

Summary

INTRODUCTION.

An elliptic quartic curve (Γ¼ for short) is the complete intersection of a (unique) pencil of quadric surfaces. We obtain an explicit description of the irreducible component H of the Hilbert scheme parametrizing all specializations of Γ¼'s.

Global descriptions of complete families of curves in projective space are seldom found in the literature. The first non trivial case, concerning the family of twisted cubics in projective 3-space, was treated by R. Piene and M. Schlessinger [PS] (cf. also [EPS] and [V],[V']). Later, a partial compactification of this family was considered by S. Kleiman, A. Strømme and S. Xambó [KSX] in order to compute characteristic numbers, and in particular to verify the number, found by Schubert, of twisted cubics tangent to 12 quadric surfaces in general position.

There is a natural rational map from the grassmannian G of pencils of quadrics to H, assigning to a pencil π its base locus β(π). The map β is not denned along the subvariety B of G consisting of pencils with a fixed component.

Let G′ denote the blowup with center B. Let β′ : G′ … H be the induced rational map; it improves the situation in the sense that the image of β′ covers now all Cohen-Macaulay curves (e.g., planar cubic union a unisecant line).

A pleasant and surprising aspect of the geometry of G′ is the appearence of the locus of indeterminacy of β as the variety of flags of doublets in a plane, denoted C′.