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 $H^n(X,E\otimes {\Omega }_X^n)$, $n=\lfloor \frac{N-1}{2}\rfloor$, 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 $d\ge 3$ and dimension at least $4$. This last statement may be viewed as a Franchetta-type conjecture for Ulrich bundles on smooth hypersurfaces.

16 pages, final version, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (to appear)