Pseudoeffective cones in 2-Fano varieties and remarks on the Voisin map

Abstract

This thesis is divided in two parts. In the first part we study the 2-Fano varieties. The 2-Fano varieties, defined by De Jong and Starr, satisfy some higher dimensional analogous properties of Fano varieties. We propose a definition of (weak) k-Fano variety and conjecture the polyhedrality of the cone of pseudoeffective k-cycles for those varieties in analogy with the case k=1. Then, we calculate some Betti numbers of a large class of k Fano varieties to prove some special case of the conjecture. In particular, the conjecture is true for all 2-Fano varieties of index > n-3, and also we complete the classification of weak 2-Fano varieties answering Questions 39 and 41 in Araujo and Castravet’s article. In the second part, we study a particular rational map. Beauville and Donagi proved that the variety of lines F(Y) of a smooth cubic fourfold Y is a hyperKähler variety. Recently, C. Lehn, M.Lehn, Sorger and van Straten proved that one can naturally associate a hyperKähler variety Z(Y) to the variety of twisted cubics on Y. Then, Voisin defined a degree 6 rational map between the direct product F(Y)xF(Y) and Z(Y). We will show that the indeterminacy locus of this map is the locus of intersecting lines.