Moduli of rank two semistable sheaves on rational Fano threefolds of the main series

TL;DR

This paper studies the moduli spaces of rank two semistable sheaves on certain rational Fano threefolds, proving boundedness results and describing their structure, including the discovery of new infinite series of components.

Contribution

It provides explicit descriptions of all moduli spaces of rank two semistable sheaves on these threefolds, including new infinite series and boundedness results for the third Chern class.

Findings

Moduli spaces are mostly irreducible smooth rational manifolds.

Identified two exceptional cases with non-rational moduli spaces.

Constructed new infinite series of rational and irrational components.

Abstract

In this paper we investigate the moduli spaces of semistable coherent sheaves of rank two on the projective space $\mathbb{P}^3$ and the following rational Fano manifolds of the main series - the three-dimensional quadric $X_2$, the intersection of two 4-dimensional quadrics $X_4$ and the Fano manifold $X_5$ of degree 5. For the quadric $X_2$, the boundedness of the third Chern class $c_3$ of rank two semistable objects in $\mathrm{D}^b(X_2)$, including sheaves, is proved. An explicit description is given of all the moduli spaces of semistable sheaves of rank two on $X_2$, including reflexive ones, with a maximal third class $c_3\ge0$. These spaces turn out to be irreducible smooth rational manifolds in all cases, except for the following two: $(c_1,c_2,c_3)=(0,2,2)$ or (0,4,8). Several new infinite series of rational components of the moduli spaces of semistable sheaves of rank two on $\mathbb{P}^3$, $X_2$, $X_4$ and $X_5$ are constructed, as well as a new infinite series of irrational components on $X_4$. The boundedness of the class $c_3$ is proved for $c_1=0$ and any $c_2>0$ for stable reflexive sheaves of general type on manifolds $X_4$ and $X_5$.