On the birational geometry of spaces of complete forms I: collineations and quadrics

TL;DR

This paper studies the birational geometry of moduli spaces of complete collineations and quadrics, computing their cones, Cox rings, and automorphism groups to understand their structure.

Contribution

It provides a detailed analysis of the Mori theory aspects of these moduli spaces, including cones, Cox rings, and chamber decompositions, which was previously unexplored.

Findings

Computed effective, nef, and movable cones.

Determined generators of Cox rings.

Described Mori chamber and base locus decompositions.

Abstract

Moduli spaces of complete collineations are wonderful compactifications of spaces of linear maps of maximal rank between two fixed vector spaces. We investigate the birational geometry of moduli spaces of complete collineations and quadrics from the point of view of Mori theory. We compute their effective, nef and movable cones, the generators of their Cox rings, and their groups of pseudo-automorphisms. Furthermore, we give a complete description of both the Mori chamber and stable base locus decompositions of the effective cone of the space of complete collineations of the 3-dimensional projective space.