Moduli of rank~$2$ semistable sheaves on~rational Fano threefolds of the main series

D. A. Vasil'ev ; A. S. Tikhomirov

Sbornik. Mathematics, Tome 215 (2024) no. 10, pp. 1269-1320

Résumé

The moduli spaces of semistable coherent sheaves of rank $2$ on the projective space $\mathbb{P}^3$ and the following rational Fano manifolds of the main series are investigated: the three-dimensional quadric $X_2$, the intersection of two four-dimensional quadrics $X_4$ and the Fano manifold of degree five $X_5$. For the quadric $X_2$ the boundedness of the third Chern class $c_3$ of rank $2$ semistable objects in $\mathrm{D}^b(X_2)$, including sheaves, is proved. An explicit description is presented for all moduli spaces of semistable sheaves of rank $2$ on $X_2$, including reflexive ones, with the maximal third class $c_3\ge0$. These spaces turn out to be irreducible smooth rational manifolds in all cases, apart from the following two: $(c_1,c_2,c_3)=(0,2,2)$ or (0,4,8). The first example of a disconnected module space of semistable rank $2$ sheaves with fixed Chern classes on a smooth projective variety is found: this is the second exceptional case $(c_1,c_2,c_3)= (0,4,8)$ on the quadric $X_2$. Several new infinite series of rational components of the moduli spaces of semistable sheaves of rank $2$ on $\mathbb{P}^3$, $X_2$, $X_4$ and $X_5$ are constructed, as also is 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 the varieties $X_4$ and $X_5$. Bibliography: 30 titles.

Détail

Keywords: stable sheaves of rank $2$, moduli spaces of sheaves, Fano manifolds.