Yeongrak Kim, Ulrich Bundles on Cubic Fourfolds

(This is a part of Algebraic Geometry Day at CCG in IBS.)

Ulrich bundles are geometric objects corresponding to maximally generated maximal Cohen-Macaulay modules, whose existence has several interesting applications in commutative algebra, homological algebra, and linear algebra. After a pioneering work of Beauville and Eisenbud-Schreyer, existence and classification of Ulrich bundles become important questions also in projective geometry. For instance, they could help to understand the cone of cohomology tables of coherent sheaves on the underlying projective variety, determinantal representations of hypersurfaces, and determinantal representations of Cayley-Chow forms. In this talk, I will discuss construction of Ulrich bundles on smooth cubic fourfolds. Unlike smooth cubic surfaces or threefolds, the smallest possible rank of Ulrich bundles on a smooth cubic fourfold may vary if it is special, i.e., X contains certain surfaces which are not homologous to complete intersections. On the other hand, a (very) general cubic fourfold does not have an Ulrich bundle of rank <6. I will explain how to construct a rank 6 Ulrich bundle on an arbitrary smooth cubic fourfold. This is a joint work with Daniele Faenzi.