Irreducibility of the moduli space of orthogonal instanton bundles on $\mathbb{P}^n$

TL;DR

This paper studies the moduli space of orthogonal instanton bundles on projective space, establishing a bijection with symmetric forms, providing explicit examples, and proving irreducibility and other geometric properties of the space.

Contribution

It introduces a correspondence between orthogonal instanton bundles and symmetric forms, and proves the irreducibility of the moduli space for certain ranks and charges.

Findings

Explicit examples of orthogonal instanton bundles without global sections.

The moduli space is affine, reduced, and irreducible when the rank reaches the upper bound.

Determination of splitting types via Kronecker modules.

Abstract

In order to obtain existence criteria for orthogonal instanton bundles on $\mathbb{P}^n$, we provide a bijection between equivalence classes of orthogonal instanton bundles with no global sections and symmetric forms. Using such correspondence we are able to provide explicit examples of orthogonal instanton bundles with no global sections on $\mathbb{P}^n$ and prove that every orthogonal instanton bundle with no global sections on $\mathbb{P}^n$ and charge $c\geq 3$ has rank $r\leq (n-1)c$. We also prove that when the rank $r$ of the bundles reaches the upper bound, $\mathcal{M}_{\mathbb{P}^n}^{\mathcal{O}}(c,r)$, the coarse moduli space of orthogonal instanton bundles with no global sections on $\mathbb{P}^n$, with charge $c\geq 3$ and rank $r$, is affine, reduced and irreducible. Last, we construct Kronecker modules to determine the splitting type of the bundles in $\mathcal{M}_{\mathbb{P}^n}^{\mathcal{O}}(c,r)$, whenever is non-empty.