Journal of the
Korean Mathematical Society JKMS

In this paper, we investigate a few strategies to construct Ulrich bundles of small ranks over smooth fourfolds in $\mathbb{P}^5$, with a focus on the case of special quartic fourfolds. First, we give a necessary condition for Ulrich bundles over a very general quartic fourfold in terms of the rank and the Chern classes. Second, we give an equivalent condition for Pfaffian fourfolds in every degree in terms of arithmetically Gorenstein surfaces therein. Finally, we design a computer-based experiment to look for Ulrich bundles of small rank over special quartic fourfolds via deformation theory. This experiment gives a construction of numerically Ulrich sheaf of rank $4$ over a random quartic fourfold containing a del Pezzo surface of degree $5$.

Keywords: Ulrich bundle, Pfaffian hypersurface, quartic fourfold

MSC numbers: Primary 14J60; Secondary 14J70, 14Q15, 13C14

Supported by: This work was supported by a 2-Year Research Grant of Pusan National University.