On the birational geometry of the parameter space for codimension 2 complete intersections
Olivier Benoist
TL;DR
The paper investigates the birational geometry of the parameter space for codimension-2 complete intersections in projective space by constructing the compactification $\bar{H}_{d_1,d_2}$ and analyzing its MMP. It proves the existence and geometric description of the first contraction and shows the full MMP can be realized in two degenerate regimes, yielding concrete birational models and a link to variation of GIT in the case $d_1=1$, $N\ge2$, as well as a detailed MMP in the punctual, $N=1$ case via a blow-up model $\hat{H}_{d_1,d_2}$ and the multigraded Hilbert scheme. These results provide a framework for constructing complete families of smooth codimension-2 complete intersections and connect to the punctual Hilbert scheme, with explicit descriptions of contractions, nef cones, and base-point-free linear systems. The work situates itself within the Hassett–Keel program for moduli spaces and interacts with prior studies on related Hilbert and GIT quotients, offering tools to probe the existence of non-isotrivial families and the structure of the associated moduli spaces.
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.