On the birational geometry of the parameter space for codimension 2 complete intersections

TL;DR

This paper investigates the birational geometry of the parameter space for codimension 2 complete intersections in projective space, describing the minimal model program and its implications for the existence of certain algebraic families.

Contribution

It provides a detailed analysis of the MMP for the parameter space of codimension 2 complete intersections, including explicit descriptions of contractions and degenerations.

Findings

First contraction of the MMP always exists and is described.

Full MMP can be run and described in two degenerate cases.

Proves existence of complete curves in the punctual Hilbert scheme of A^2.

Abstract

Codimension 2 complete intersections in P^N have a natural parameter space \bar{H}: a projective bundle over a projective space given by the choice of the lower degree equation and of the higher degree equation up to a multiple of the first. Motivated by the question of existence of complete families of smooth complete intersections, we study the birational geometry of \bar{H}. In a first part, we show that the first contraction of the MMP for \bar{H} always exists and we describe it. Then, we show that it is possible to run the full MMP for \bar{H}, and we describe it, in two degenerate cases. As an application, we prove the existence of complete curves in the punctual Hilbert scheme of complete intersection subschemes of A^2.