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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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