On the birational geometry of spaces of complete forms II: Skew-forms

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.

On the Weak Lefschetz Principle in Birational Geometry

“…Recently in [12][13][14][15] the birational geometry of classical spaces, called spaces of complete forms, have been studied from the perspective of Definition 4. We finish up the article by mentioning some of the ingredients of such a study.…”

“…Example 2. The space (3) is the blow-up of the projective space ℙ 15 along the Segre variety ≅ ℙ 3 × ℙ 3 ⊂ ℙ 15 , and also along the strict transform of sec 2 ( ). We will denote by the strict transform of a general hyperplane of ℙ 15 and by 1 , 2 the exceptional divisors over and sec 2 ( ), respectively.…”

Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming

“…Just as with complete quadrics, there are many ways of constructing this space. Here we give just two, referring the reader to the literature [2,41,21] for other equivalent definitions. Definition 7.4.…”

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

“…Anyway, whenever some additional argument is needed we will explain it in detail. While this paper is devoted to spaces of complete collineations and quadrics, their skew‐symmetric counterpart, that is the space of complete skew‐symmetric forms, is considered in [36].…”