Toric Fano varieties and birational morphisms
Cinzia Casagrande

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 , for the difference of the Picard numbers of X and D. Moreover, if (with some additional hypotheses if ), we give an explicit birational description of X. Using this result, we show that when dim X=5, we have . 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.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Algebraic Geometry and Number Theory · Algebraic structures and combinatorial models
