The moduli space of torsion-free sheaves with quasi-maximal third Chern class

Abstract

In this paper, we investigate the geometry of the Gieseker moduli space of semistable rank $2$ torsion-free sheaves on $\mathbb{P}^3$ with Chern classes $(c_1, c_2, c_3) = (-1, c_2, c_2^2-2)$ for $c_2 \geq 2$. We use the modular Serre correspondence to relate these moduli spaces to spaces of pairs and to suitable Hilbert schemes of one-dimensional subschemes in $\mathbb{P}^3$. For $c_2 \geq 4$, we prove that the moduli space is irreducible of dimension $c_2^2+3c_2+5$. For the case $c_2 = 3$, relying on a known geometric description of a relevant Hilbert scheme, we show that the moduli space $\mathcal{M}(-1,3,7)$ consists of exactly two irreducible components, namely the generic component of reflexive sheaves and a $T$-component, and we prove that their intersection is nonempty.