Homological Projective Duality (HPD) is a rather powerful theory that allows to reason about derived categories of coherent sheaves. In its simplest form, it begins with a smooth projective variety and a rather nice semiorthogonal decomposition of its bounded derived category (a so-called Lefschetz decomposition). According to HPD, one can find another varitey (often nocommutative) with a map to the dual projective spaces and its own Lefschetz decomposition such that the derived categories of the complimentary hyperplane sections of the two are related in a very explicit way. Among other things, HPD has shown over the years to be an very elegant way to construct embeddings and equivalences between derived categories of classical varieties.
It is very natural to study how HPD behaves with respect to classical geometric constructions. In their previous paper, Categorical joins, for a pair of projective vatieties with HPD duals, the authors constructed the HPD dual of their join (more precisely, an noncommutative resolution of the classical join) equipped with a natural Lefschetz semiorthogonal decomposition. In the present paper the authors concentrate on a categorical version of classical geometric cones. Let \(C(X)\) be a cone over a smooth projective variety \(X\). While \(C(X)\) is singular, it has a natural resolution given by the blow up of its vertex. From the point of view of derived categories, this resolution is, as it often happens, too big. The authors define the categorical cone as a certain triangulated subcategory of the blowup. Moreover, they are able to do in a very general setting (when \(X\) is a noncommuataive variety). Given some mild technical assumptions, which are satisfied in practice, the authors prove an HPD theorem relating the dual of a variety to the dual of a cone over it.
HPD for cones is next applied to the study of HPD for singular quadrics. Finally the authors give applications to the study of Gushel-Mukai varieties. According to previously obtained results, to every Gushel-Mukai variety one can associate a triangulated category which is either a noncommutative \(K3\) or Enriques surface, depending on the parity of its dimension. It is expected that if two GM-varieties of the same dimension have the same noncommutative surface category, then they are isomophic. The authors settle a certain duality conjecture, which gives strong evidence to the previous statement (we refer the reader to Section 1.5 for details).