Classification of monads and moduli components of stable rank 2 bundles with odd determinant and $c_2=10$

TL;DR

This paper classifies certain stable rank 2 bundles on projective 3-space with specific Chern classes, introduces a new moduli component, and characterizes spectra of these bundles using minimal monads and Rao modules.

Contribution

It provides a complete classification of positive minimal monads for these bundles, proves the existence of a new irreducible moduli component, and characterizes spectra via Hartshorne's conditions.

Findings

Existence of a new irreducible component of the moduli space.

Complete classification of positive minimal monads for the bundles.

Characterization of spectra satisfying Hartshorne's conditions.

Abstract

In this paper, we provide a complete classification of the positive minimal monads whose cohomology is a stable rank 2 bundle on $\mathbb{P}^3$ with Chern classes $c_1=-1, c_2=10$ and we prove the existence of a new irreducible component of the moduli space $\mathcal{B}(-1,10)$ of a rank 2 stable bundles with the given Chern classes. We also show that Hartshorne's conditions on a sequence $\mathcal{X}$ of 10 integers are sufficient and necessary for the existence of a stable rank 2 bundle with odd determinant and spectrum $\mathcal{X}$. Furthermore, we prove that the sequence of integers $\{-2^{n-1},-1,0,1^{n-1}\}$ for $ n\geq4$ is realized as the spectrum of a stable rank 2 bundle $\EE$ of odd determinant by computing the minimal generators of its Rao module.