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2 papers
Moduli of rank $2$ semistable sheaves on rational Fano threefolds of the main series
D. A. Vasil'eva,
A. S. Tikhomirovb a Laboratory of Algebraic Geometry and Its Applications, National Research University Higher School of Economics, Moscow, Russia
b Faculty of Mathematics, National Research University Higher School of Economics, Moscow, Russia
Abstract:
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.
Keywords:
stable sheaves of rank $2$, moduli spaces of sheaves, Fano manifolds.
MSC: 14D20,
14F06,
14F45,
14J60 Received: 19.02.2024 and 01.07.2024
DOI:
10.4213/sm10087
Fulltext:
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