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

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 three-dimensional projective space. Contents

Complete singular collineations and quadrics

“…The case Cpn, m, n `1q and Qpn, n `1q are respectively the space of complete collineations from V to W and the space of complete quadrics of V . By [Vai84, Theorem 1] and [Vai82, Theorem 6.3] they are wonderful varieties and their birational geometry has been studied in [Mas20a].…”

“…Complete collineations have been widely studied from the algebraic, enumerative and birational viewpoint since the 19th-century [Cha64], [Gia03], [Hir75], [Hir77], [Sch86], [Seg84], [Sem48], [Sem51], [Sem52], [Tyr56], [Vai82], [Vai84], [KT88], [LLT89], [Tha99], [Hue15], [Mas20a], [Mas20b].…”

“…The automorphism group of M 0,0 pP n , 2q for n ě 3 follows from Proposition 4.1 and Theorem 3.21. The automorphism group of M 0,0 pP 2 , 2q has been computed in [Mas20a,Remark 7.6]. Finally, to get the claim on AutpM 0,0 pP 1 , 2qq notice that M 0,0 pP 1 , 2q -P 2 .…”

Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming

“…The space of complete quadrics CQ(V ) has several equivalent descriptions, below we will describe some of them. For more information we refer the reader to [17,41,20].…”

“…What is the number of nondegenerate quadrics in n variables, passing through In modern language, such problems can be solved by performing computations in the cohomology ring of the variety of complete quadrics. This is now a classical topic with many beautiful results [32,33,42,43,8,9,17,7,41,20]. In particular, the cohomology ring has been described by generators and relations, and algorithms have been devised that allow to compute any given intersection number.…”