Toric Fano varieties and birational morphisms

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 $0\le {\rho }_X-{\rho }_D\le 3$, 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 ${\rho }_X\le 9$. 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.

LaTeX, 35 pages, 27 figures, 2 tables