Toric Fano varieties and birational morphisms

2001/12/01 by Cinzia Casagrande, Casagrande, Cinzia
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG

paper · pdf · doi:10.48550/arxiv.math/0112007

LaTeX, 35 pages, 27 figures, 2 tables

arxiv created 2001/12/01 · arxiv updated 2009/11/30

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≤ρ_X-ρ_D≤ 3, for the difference of the Picard numbers of X and D. Moreover, if ρ_X-ρ_D>0 (with some additional hypotheses if ρ_X-ρ_D=1), we give an explicit birational description of X. Using this result, we show that when dim X=5, we have ρ_X≤ 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.

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