Construction of stable rank 2 vector bundles on $\mathbb{P}^3$ via symplectic bundles

TL;DR

This paper constructs new infinite series of irreducible components in the moduli space of stable rank 2 vector bundles on projective 3-space, using symplectic bundles and monads, expanding understanding of their structure.

Contribution

It introduces two new infinite series of irreducible components in the moduli space, constructed via cohomology sheaves of monads involving symplectic bundles.

Findings

Series  contains components for all large n (n6).

Provides new examples of infinite series of components in the moduli space.

Uses symplectic instanton and twisted symplectic bundles in construction.

Abstract

In this article we study the Gieseker-Maruyama moduli spaces $\mathcal{B}(e,n)$ of stable rank 2 algebraic vector bundles with Chern classes $c_1=e\in\{-1,0\},\ c_2=n\ge1$ on the projective space $\mathbb{P}^3$. We construct two new infinite series $\Sigma_0$ and $\Sigma_1$ of irreducible components of the spaces $\mathcal{B}(e,n)$, for $e=0$ and $e=-1$, respectively. General bundles of these components are obtained as cohomology sheaves of monads, the middle term of which is a rank 4 symplectic instanton bundle in case $e=0$, respectively, twisted symplectic bundle in case $e=-1$. We show that the series $\Sigma_0$ contains components for all big enough values of $n$ (more precisely, at least for $n\ge146$). $\Sigma_0$ yields the next example, after the series of instanton components, of an infinite series of components of $\mathcal{B}(0,n)$ satisfying this property.