We prove a vanishing theorem for the Hodge number h 2;1 of projective toric varieties provided by a certain class of polytopes. We explain how this Hodge number also gives information about the deformation theory of the toric Gorenstein singularity derived from the same polytope. In particular, the vanishing theorem for h 2;1 implies that these deformations are unobstructed.

We prove that Vopěnka's Principle implies for every class X of modules over any ring, the X-Gorenstein Projective (X-GP) is a precovering class. In particular, it not possible to (unless inconsistent) there ring which Ding Projectives (DP) or Gorenstein (GP) do form (Šaroch previously obtained this result GP, using different methods). The key innovation new “top-down” characterization deconstru...

Let [Formula: see text] be an abelian category. In this paper, we investigate the global text]-Gorenstein projective dimension text], associated to a GP-admissible pair text]. We give homological conditions over that characterize it. Moreover, given GI-admissible study under which and are same.

In the present note we investigate projective varieties which are geometric models of binary symmetric phylogenetic 3-valent trees. We prove that these varieties have Gorenstein terminal singularities (with small resolution) and they are Fano varieties of index 4. Moreover any two such varieties associated to trees with the same number of leaves are deformation equivalent, that is, they are in ...

In [Pas15], we described the Minimal Model Program in the family of Q-Gorenstein projective horospherical varieties, by studying a family of polytopes defined from the moment polytope of an ample Q-Cartier Q-divisor of the variety we begin with. Here, we summarize the results of [Pas15] and we explain how to generalize them in order to describe the Log Minimal Model Program for pairs (X,∆) wher...