Deformations of half-canonical Gorenstein curves in codimension four (Q6051118)

Despite the extensive literature on Gorenstein rings, many fundamental questions in this fascinating topic are still open, such as explicit constructions in codimension four and beyond. The paper under review addresses the study of Gorenstein projective varieties for codimension four from a computational point of view. More precisely, the authors focus on Gorenstein algebras of codimension and regularity four defining varieties. A description of all possible Betti tables for these varieties is already available in terms of Artinian Gorenstein algebras, thanks to previous papers by several authors. The interest for the regularity four arises from its necessity for Gorenstein Calabi-Yau threefolds in \(\mathbb P^7\). Some of the Betti tables occur for at least one family of non-singular threefolds in \(\mathbb P^7\), while others cannot occur for non-singular threefolds. The authors of this paper show that some of the singular varieties can be smoothed to one of the non-singular varieties by suitable flat deformations, hence in the same Hilbert scheme (see Theorem 1.2).