Instanton bundles on $\mathbb{P}^1\times\mathbb{F}_1$

TL;DR

This paper studies rank two instanton bundles on the Fano threefold _1 _1 imes \u00a7^1, providing a monad description, proving their existence for various Chern classes, and analyzing their moduli spaces and properties.

Contribution

It introduces a monad construction for instanton bundles on _1 _1, proves their existence for all admissible Chern classes, and characterizes minimal instantons as arithmetically Cohen-Macaulay.

Findings

Every instanton bundle can be described as the cohomology of a monad.

Existence of instanton bundles for any admissible second Chern class.

Minimal instanton bundles are arithmetically Cohen-Macaulay and their moduli space is described.

Abstract

In this paper we deal with a particular class of rank two vector bundles (\emph{instanton} bundles) on the Fano threefold of index one $F:=\mathbb{F}_1 \times \mathbb{P}^1$. We show that every instanton bundle on $F$ can be described as the cohomology of a monad whose terms are free sheaves. Furthermore we prove the existence of instanton bundles for any admissible second Chern class and we construct a nice component of the moduli space where they sit. Finally we show that minimal instanton bundles (i.e. with the least possible degree of the second Chern class) are aCM and we describe their moduli space.