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

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 $c_1=e\in \{-1,0\},c_2=n\ge 1$ 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 $n\ge 146$). ${\Sigma }_0$ yields the next example, after the series of instanton components, of an infinite series of components of $\mathcal{B}(0,n)$ satisfying this property.

17 pages