1. Hodge rank of ACM bundles and Franchetta's conjecture
- Author
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Biswas, Indranil and Ravindra, G. V.
- Subjects
Mathematics - Algebraic Geometry ,14J60, 14D21 - 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., Comment: 16 pages, final version, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (to appear)
- Published
- 2023