Yeongrak Kim
Pusan National University, Busan, South Korea

An Ulrich sheaf on a closed subscheme $X$ of $\mathbb{P}^N$ of dimension $n$ and degree $d$ is a nonzero coherent sheaf $\mathcal F$ on $X$ whose cohomology table $\{h^i (X, \mathcal F (j))\}$ is a multiple of the cohomology table of the structure sheaf of $\mathbb{P}^n$. When $X=V(F)$ is a hypersurface, studying Ulrich sheaves on $X$ is closely related to determinantal representations of $F$ and matrix factorizations of $F$ in the sense of Eisenbud. In this talk, we discuss the existence of rank $6$ Ulrich bundles on a smooth cubic fourfold. The main idea is to construct a sheaf which has the same topological type as an Ulrich bundle of given rank using the twisted cubics lying on a cubic fourfold, and then to deform it into a locally free sheaf. This is a joint work with D. Faenzi.