On the birational geometry of toric Fano [formula omitted]-folds

Abstract

In this Note, we announce a factorization result for equivariant birational morphisms between toric 4-folds whose source is Fano: such a morphism is always a composite of blow-ups along smooth invariant centers. Moreover, we show with a counterexample that, differently from the three-dimensional case, even if both source and target are Fano, the intermediate varieties can not be chosen Fano.