Even and odd instanton bundles on Fano threefolds

TL;DR

This paper introduces non-ordinary instanton bundles on Fano threefolds, establishing their existence, describing their moduli spaces, and analyzing their properties and restrictions, including on projective space and quadrics.

Contribution

It extends the concept of instanton bundles to non-ordinary cases on Fano threefolds, providing bounds, moduli space descriptions, and monadic representations.

Findings

Existence of non-ordinary instanton bundles for all admissible quantum numbers when certain conditions hold.

Identification of components in moduli spaces containing these bundles.

Description of loci of jumping lines and monadic representations on specific varieties.

Abstract

We define non-ordinary instanton bundles on Fano threefolds $X$ extending the notion of (ordinary) instanton bundles. 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 $i_X\ge 2$ or $i_X=1$, $\mathrm{Pic}(X)$ is cyclic and $X$ is ordinary. 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.