Two infinite series of moduli spaces of rank 2 sheaves on $\mathbb{P}^3$

TL;DR

This paper investigates new components of the moduli space of rank 2 sheaves on projective 3-space, revealing their growth, intersections, and the connectedness of certain moduli spaces, with implications for understanding their structure.

Contribution

It introduces new components of the moduli space of rank 2 sheaves on , analyzes their growth and intersections, and proves connectedness results for specific cases.

Findings

Number of components grows with n

Identifies intersection patterns with the instanton component

Proves  is connected and describes its components

Abstract

We describe new components of the Gieseker--Maruyama moduli scheme $\mathcal{M}(n)$ of semistable rank 2 sheaves $E$ on $\mathbb{P}^3$ with $c_1(E)=0$, $c_2(E)=n$ and $c_3(E)=0$ whose generic point corresponds to non locally free sheaves. We show that such components grow in number as $n$ grows, and discuss how they intersect the instanton component. As an application, we prove that $\mathcal{M}(2)$ is connected, and identify a connected subscheme of $\mathcal{M}(3)$ consisting of 7 irreducible components.