Toric Fano varieties and birational morphisms

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