Toric Fano Varieties and Birational Morphisms

Smooth toric Fano varieties are classified up to dimension 4. In dimension 2, there are five toric Del Pezzo surfaces: P, P1×P1, and Si, the blowup of P in i points, for i = 1, 2, 3. There are 18 toric Fano 3-folds [2, 20] and 124 toric Fano 4-folds [4, 17]. In higher dimensions, little is known about them and many properties that hold in low dimensions are not known to hold in general. Let X be a toric Fano variety of dimension n. We recall that in X, linear and algebraic equivalence for divisors coincide; so the Picard number ρX is the rank of the Picard group of X. By means of some basic combinatorial properties of ΣX, we show that for every irreducible invariant divisorD ⊂ X,we have 0 ≤ ρX−ρD ≤ 3. Moreover, if ρX−ρD = 3, then X is a toric S3-bundle over a lower-dimensional toric Fano variety; if 1 ≤ ρX−ρD ≤ 2 (with some additional hypotheses in the case ρX−ρD = 1), we give an explicit birational description of X. This gives a structure theorem for a large class of toric Fano varieties (Theorem 3.4). The best known bound for the Picard number of X is, due to Debarre [8], ρX ≤ 2 + 2 √ (n2 − 1)(2n− 1). On the other hand, we know that the maximal Picard number in dimension n is at least 2n for n even, 2n − 1 for n odd. In fact, up to dimension 4, this is exactly the maximal Picard number. We show that the same is true in dimension 5: if X is a toric Fano 5-fold, ρX ≤ 9 (Theorem 4.2). In the second part of the paper, we study equivariant morphisms f : X → Y whose source X is Fano. As a direct application of Theorem 3.4, we show that for every irreducible invariant divisor D ⊂ X, we have ρY − ρf(D) ≤ 3 (Proposition 5.1).