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Toric Fano varieties and birational morphisms. (English) Zbl 1083.14516
Summary: 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 $$0\leq\rho (X)-\rho (D)\leq3$$ 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 $$\rho (X)\leq 9$$. Then, we study equivariant morphisms whose source is Fano. We give some general results, and in dimension 4, we show that a birational equivariant morphism is always a composite of smooth equivariant blowups. Finally, we study under which hypotheses a nonprojective toric variety can become Fano after a smooth equivariant blowup.

##### MSC:
 14J25 Special surfaces 14E05 Rational and birational maps 14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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