Asian Journal of Mathematics

Volume 26 (2022)

Number 1

Even and odd instanton bundles on Fano threefolds

Pages: 81 – 118

DOI: https://dx.doi.org/10.4310/AJM.2022.v26.n1.a4

Authors

Vincenzo Antonelli (Dipartimento di Scienze Matematiche, Politecnico di Torino, Italy)

Gianfranco Casnati (Dipartimento di Scienze Matematiche, Politecnico di Torino, Italy)

Ozhan Genc (Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland)

Abstract

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