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Moduli of simple rank-2 sheaves on \(K3\)-surfaces. (English) Zbl 0812.14024

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 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 \(\text{Spl} (c_ 1, c_ 2)\) such that no sheaf in \({\mathcal M}\) is stable. Then,
(i) \({\mathcal M}\) is birational to either \(\text{Hilb}^ d (X)\) or \(X \times \text{Hilb}^{d - 1}(X)\);
(ii) there exists a divisor \(F\) with \((c_ 1 - 2F)^ 2 = (4c_ 2 - c^ 2_ 1) - 12\).
This is proved in section two. – In section three, we give examples of those divisors \(F\) in
theorem B (ii). \(\text{Spl} (c_ 1, c_ 2)\) may contain infinitely many irreducible components:
Theorem C. Assume that \(c\) is an odd integer larger than three, and that \(X\) is an elliptic \(K3\)-surface. Then, \(\text{Spl} (0,c)\) contains infinitely many irreducible components if and only if the Picard number of \(X\) is larger than two.
On the other hand, if \(c\) is even and larger than four, then \(\text{Spl} (0,c)\) has finitely many irreducible components. Finally, in section four, we study \({\mathcal M}_ L (c_ 1, c_ 2)\) for an elliptic \(K3\)- surface and for arbitrary \(c_ 1\).
Theorem D. Let \(X\) be an elliptic \(K3\)-surface such that any fiber of \(j\) is irreducible and has at worst ordinary double points as singularities. If the moduli space \({\mathcal M}_ L (c_ 1,c_ 2)\) is nonempty, then it is birational to \(\text{Hilb}^ d (X)\).
This result, together with theorem B and theorem C, gives a complete answer to problem A when \(X\) is an elliptic \(K3\)-surface.

MSC:

14J10 Families, moduli, classification: algebraic theory
14J28 \(K3\) surfaces and Enriques surfaces
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
14D20 Algebraic moduli problems, moduli of vector bundles
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14C05 Parametrization (Chow and Hilbert schemes)

Citations:

Zbl 0631.14009
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References:

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