Toric Fano varieties and birational morphisms

TL;DR

This paper investigates the structure of smooth toric Fano varieties, providing bounds on Picard numbers, explicit birational descriptions, and conditions under which non-projective varieties become Fano after blow-ups.

Contribution

It offers new bounds on Picard numbers related to invariant divisors and characterizes birational morphisms and blow-ups in the context of toric Fano varieties.

Findings

For any invariant divisor D, 0 ≤ ρ_X - ρ_D ≤ 3.

In dimension 5, the Picard number ρ_X ≤ 9.

In dimension 4, equivariant birational morphisms are compositions of smooth blow-ups.

Abstract

In this paper we study smooth toric Fano varieties using primitive relations and toric Mori theory. We show that for any irreducible invariant divisor D in a toric Fano variety X, we have $0\leq\rho_X-\rho_D\leq 3$, for the difference of the Picard numbers of X and D. Moreover, if $\rho_X-\rho_D>0$ (with some additional hypotheses if $\rho_X-\rho_D=1$), we give an explicit birational description of X. Using this result, we show that when dim X=5, we have $\rho_X\leq 9$. In the second part of the paper, we study equivariant birational morphisms f whose source is Fano. We give some general results, and in dimension 4 we show that f is always a composite of smooth equivariant blow-ups. Finally, we study under which hypotheses a non-projective toric variety can become Fano after a smooth equivariant blow-up.