Monads and moduli components for stable rank 2 bundles with odd determinant on the projective space

TL;DR

This paper advances the classification of stable rank 2 vector bundles with odd determinant on projective 3-space, identifying new moduli space components and providing a comprehensive analysis for bundles with second Chern class up to 8.

Contribution

It offers a complete classification of such bundles with positive minimal monads and describes new irreducible components of their moduli spaces.

Findings

Classified all stable rank 2 bundles with odd determinant and c2 ≤ 8.

Identified all spectra and minimal monads for these bundles.

Disproved a question by Hartshorne and Rao.

Abstract

We propose a three-step program for the classification of stable rank 2 bundles on the projective space $\mathbb{P}^3$ inspired by an article by Hartshorne and Rao. While this classification program has been successfully completed for stable rank 2 bundles with even determinant and $c_2\le5$, much less is known for bundles with odd determinant. After revising the known facts about these objects, we list all possible spectra and minimal monads for stable rank 2 bundles with odd determinant and $c_2\le8$. We provide a full classification of all bundles with positive minimal monads, provide a negative answer to a question raised by Hartshorne and Rao, and describe new irreducible components of the moduli spaces of stable rank 2 bundles with odd determinant and $c_2=6,8$.