Séminaire Francilien de Géométrie Algorithmique et Combinatoire

15h30 Mathieu Vallée Université Libre de Bruxelles

Complete nonsingular toric varieties of Picard number 4

Toric varieties form a specific class of algebraic varieties equipped with a well-behaved action of an algebraic torus. They provide a useful setting for testing conjectures, as they admit a particularly explicit combinatorial description. The fundamental theorem of toric geometry states that toric varieties correspond to fans, that is, collections of strongly convex polyhedral cones in R^n that are closed under taking faces and whose interiors are pairwise disjoint. Properties of the fan translate directly into geometric properties of the associated toric variety. In particular, a toric variety is complete if and only if the cones of the fan cover the whole space R^n , and it is non-singular if and only if each cone is generated by part of a basis of the integer lattice Z^n . We focus here on characterizing complete non-singular toric varieties, also called toric manifolds. The Picard number of a complete fan is the number of its 1-dimensional cones minus the dimension n; this equals the rank of the Picard group of the associated toric manifold. There are two major directions of research: studying toric manifolds of fixed (small) dimension, or studying those with fixed (small) Picard number. In dimension 2, toric manifolds are completely understood: they are obtained from either the complex projective plane or a Hirzebruch surface by a sequence of toric blow-ups. In any dimension, the unique toric manifold of Picard number 1 is the complex projective space CP^n, whose fan corresponds to the normal fan of a unimodular simplex of dimension n. Kleinschmidt (1988) and Batyrev (1991) classified toric manifolds of Picard number 2 and 3, respectively. In this talk, I will present a sequence of joint works with Suyoung Choi and Hyeontae Jang leading to the classification of toric manifolds of Picard number 4 in terms of fans, realying mainly on a combinatorial construction known as the (simplicial) wedge operation.