Moduli spaces of rank 2 instanton sheaves on the projective space

Marcos Jardim, Mario Maican and Alexander S. Tikhomirov

Vol. 291 (2017), No. 2, 399–424

DOI: 10.2140/pjm.2017.291.399

Abstract

We study the irreducible components of the moduli space of instanton sheaves on ${\mathbb{P}}^3$, that is, $\mu$-semistable rank 2 torsion-free sheaves $E$ with $c_1\left(E\right)=c_3\left(E\right)=0$ satisfying $h^1\left(E\left(-2\right)\right)=h^2\left(E\left(-2\right)\right)=0$. In particular, we classify all instanton sheaves with $c_2\left(E\right)\le 4$, describing all the irreducible components of their moduli space. A key ingredient for our argument is the study of the moduli space $\mathcal{T}\left(d\right)$ of stable sheaves on ${\mathbb{P}}^3$ with Hilbert polynomial $P\left(t\right)=d\cdot t$, which contains, as an open subset, the moduli space of rank 0 instanton sheaves of multiplicity $d$; we describe all the irreducible components of $\mathcal{T}\left(d\right)$ for $d\le 4$.

Keywords

Moduli spaces, semistable sheaves, instanton sheaves, sheaves on projective space

Mathematical Subject Classification 2010

Primary: 14D20, 14J60

Milestones

Received: 23 February 2017

Revised: 10 May 2017

Accepted: 26 May 2017

Published: 14 September 2017