Instanton bundles on the Segre threefold with Picard number three

TL;DR

This paper investigates instanton bundles on the Segre threefold with Picard number three, constructing monads, characterizing stability, and analyzing jumping lines to deepen understanding of their geometric properties.

Contribution

It introduces new monad constructions for instanton bundles on the Segre threefold and characterizes their stability and jumping line loci.

Findings

Existence of $ ext{μ}$-stable instanton bundles for all $c_2(E)$

Characterization of Gieseker semistability conditions

Description of the jumping lines locus

Abstract

We study instanton bundles $E$ on $\mathbb{P}^1\times \mathbb{P}^1 \times \mathbb{P}^1$. We construct two different monads which are the analog of the monads for instanton bundles on $\mathbb P^3$ and on the flag threefold $F(0,1,2)$. We characterize the Gieseker semistable cases and we prove the existence of $\mu$-stable instanton bundles generically trivial on the lines for any possible $c_2(E)$. We also study the locus of jumping lines.