Complete non-singular 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 and combinatorial description. The fundamental theorem of toric geometry states that toric varieties correspond to fans, that is, collections of strongly convex polyhedral cones in ℝⁿ that are closed under taking faces and whose relative 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 $ℝⁿ$, and it is non-singular if and only if each cone is generated by part of a basis of the integer lattice $ℤⁿ$ . We focus here on characterizing complete non-singular toric varieties, also called toric manifolds. The Picard number of a toric manifold is the rank of its Picard group; this equals the number of the 1-dimensional cones minus the dimension n of its associated fan. There are two major directions of research toward this characterization: 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 by a sequence of toric blow-ups startying either from the complex projective plane or from a Hirzebruch surface. In any dimension n, the unique toric manifold of Picard number 1 is the complex projective space $ℂPⁿ$, whose fan corresponds to the normal fan of a unimodular n-simplex. 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, relying mainly on a combinatorial construction known as the (simplicial) wedge operation; which was used for instance by F. Santos in his construction for disproving the Hirsch conjecture.