Moduli of simple rank-2 sheaves on \(K3\)-surfaces
type: Item
instance of: scholarly article
title: Moduli of simple rank-2 sheaves on \(K3\)-surfaces
zbMATH Open document ID: 0812.14024
DOI: 10.1007/BF02568344
author: Zhenbo Qin
published in: Manuscripta Mathematica
publication date: 1995-05-14T00:00:00Z
full work available at URL: https://eudml.org/doc/155840
review text: Let \(X\) be an algebraic surface. Fix a divisor \(c_ 1\), an integer \(c_ 2\), and a polarization \(L\) on \(X\). Let \(\text{Spl} (c_ 1, c_ 2)\) be the moduli space of simple torsion-free rank-2 sheaves with chern classes \(c_ 1\) and \(c_ 2\), and let \({\mathcal M}_ L (c_ 1,c_ 2)\) be the Zariski open subset of \(\text{Spl} (c_ 1, c_ 2)\) consisting of \(L\)- stable locally free sheaves. For sufficiently large \((4c_ 2-c^ 2_ 1)\), Donaldson and Friedman showed that \({\mathcal M}_ L( c_ 1, c_ 2)\) is nonempty and generically smooth with dimension \((4c_ 2 - c^ 2_ 1 - 3_ \chi ({\mathcal O}_ X))\). For a \(K3\)-surface \(X\), by a well-known result of Mukai, if \(\text{Spl} (c_ 1, c_ 2)\) is nonempty, then it is smooth with dimension \((4c_ 2 - c^ 2_ 1 - 6)\) and has a symplectic structure, i.e., a nowhere-degenerate holomorphic form. Put \(d = (4c_ 2 - c_ 1^ 2 - 6)/2\), and let \(\text{Hilb}^ d(X)\) be the Hilbert scheme parametrizing all 0-cycles of \(X\) with length \(d\). In this paper, we solve the following problem (rank-2 case) raised by \textit{A. N. Tyurin} [cf. Duke Math. J. 54, 1-26 (1987; Zbl 0631.14009)]. Problem A. Assume that the moduli space \(\text{Spl} (c_ 1, c_ 2)\) is nonempty. (i) What is the birational structure of \(\text{Spl} (c_ 1, c_ 2)\)? (ii) Is \(\text{Spl} (c_ 1, c_ 2)\) birational to the Hilbert scheme \(\text{Hilb}^ d(X)\)? Indeed, by Fujiki, Beauville and Mukai, \(\text{Hilb}^ d(X)\) admits a symplectic structure. If \((4c_ 2 - c^ 2_ 1) \geq 12\) and if an irreducible component \({\mathcal M}\) of \(\text{Spl} (c_ 1, c_ 2)\) contains a stable sheaf, then it contains a locally free sheaf; thus, an open subset of \({\mathcal M}\) is contained in \({\mathcal M} (c_ 1, c_ 2)\). Our first result gives the birational structures of those irreducible components which contain no stable sheaf. Theorem B. Let \(X\) be a \(K3\)-surface. Assume that \((4c_ 2 - c^ 2_ 1) > 16\), and that \({\mathcal M}\) is an irreducible component in \(\text{Spl…
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14J27
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14C05
zbMATH DE Number: 492746
zbMATH Keywords: elliptic \(K3\)-surface
zbMATH Keywords: moduli space
zbMATH Keywords: Hilbert scheme
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 492746