Hodge rank of ACM bundles and Franchetta's conjecture

Abstract

We prove that on a general hypersurface in $\mathbb{P}^N$ of degree $d$ and dimension at least $2$, if an arithmetically Cohen-Macaulay (ACM) bundle $E$ and its dual have small regularity, then any non-trivial Hodge class in $H^{n}(X, E\otimes\Omega^n_X)$, $n = \lfloor\frac{N-1}{2}\rfloor$, produces a trivial direct summand of $E$. As a consequence, we prove that there is no universal Ulrich bundle on the family of smooth hypersurfaces of degree $d\geq 3$ and dimension at least $4$. This last statement may be viewed as a Franchetta-type conjecture for Ulrich bundles on smooth hypersurfaces.