Even and odd instanton bundles on Fano threefolds

We define non–ordinary instanton bundles on Fano threefolds $X$ extending the notion of (ordinary) instanton bundles introduced in [14]. We determine a lower bound for the quantum number of a non–ordinary instanton bundle, i.e. the degree of its second Chern class, showing the existence of such bundles for each admissible value of the quantum number when $i_X \geq 2$ or $i_X = 1$ and $\mathrm{rk} \: \operatorname{Pic}(X) = 1$. In these cases we deal with the component inside the moduli spaces of simple bundles containing the vector bundles we construct and we study their restriction to lines. Finally we give a monadic description of non–ordinary instanton bundles on $\mathbb{P}^3$ and the smooth quadric studying their loci of jumping lines, when of the expected codimension.

Keywords

Fano threefold, vector bundle, $\mu$-(semi)stable bundle, simple bundle, instanton bundle

2010 Mathematics Subject Classification

Primary 14J60. Secondary 14D21, 14J30, 14J45.

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The first- and second-named authors are members of the GNSAGA group of INdAM and are supported by the framework of the MIUR grant Dipartimenti di Eccellenza 2018-2022 (E11G18000350001).

The third-named author is supported by the grant MAESTRO NCN-UMO-2019/34/A/ST1/00263, “Research in Commutative Algebra and Representation Theory”.

Received 14 July 2020

Accepted 9 September 2021

Published 30 January 2023