Hodge rank of ACM bundles and Franchetta's conjecture

Indranil Biswas; G. V. Ravindra

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arXiv · 2306.03587

Hodge rank of ACM bundles and Franchetta's conjecture

Indranil Biswas·G. V. Ravindra

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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.

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BibTeX RIS

Indranil Biswas, G. V. Ravindra. 2023-06-06. Hodge rank of ACM bundles and Franchetta's conjecture. https://arxiv.org/abs/2306.03587

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