We study Enriques surfaces with four disjoint A2-configurations. In particular, we construct open Enriques surfaces with fundamental groups (Z/3Z) × Z/2Z and Z/6Z, completing the picture of the A2-case from [10]. We also construct an explicit Gorenstein Q-homology projective plane of singularity type A3 + 3A2, supporting an open case from [7].

We prove a vanishing theorem for the Hodge number h 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 implies that these deformations are unobstructed

Let X ⊂ P be a generically reduced projective scheme. A fundamental goal in computational algebraic geometry is to compute information about X even when defining equations for X are not known. We use numerical algebraic geometry to develop a test for deciding if X is arithmetically Gorenstein and apply it to three secant varieties.

We recall the basic geometric properties of the projective variety Latn r (K) parametrizing a family of special lattices over Witt vectors proved in [Hab05, HS, San04]. In this paper, we prove that a particular set of subvarieties of Latn r (K) are normal and Gorenstein. The set contains the subregular variety, that is, the complement of the smooth locus, of Latn r (K).

The so-called ’change-of-ring’ results are well-known expressions which present several connections between projective, injective and flat dimensions over the various base rings. In this note we extend these results to the Gorenstein dimensions over Cohen-Macaulay local rings.