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Fourier-Mukai transforms for elliptic surfaces. (English) Zbl 0905.14020
This paper introduces a new class of generalized Fourier-Mukai transforms. These take the form of equivalences between derived categories of sheaves on certain pairs of algebraic elliptic surfaces. As an application a large number of moduli spaces of stable sheaves are calculated, generalising work of R. Friedman.
Let \(\pi: X\to C\) be a simply-connected, complex algebraic elliptic surface, and let \(f\) denote the divisor class of a fibre of \(\pi\). Then one can construct a new elliptic surface \(\widehat\pi: J_X(a,b)\to C\) whose fibre over a point \(p\in C\) is naturally identified with the space of stable sheaves of rank \(a\) and degree \(b\) on the fibre \(\pi^{-1} (p)\). Here \(a>0\) and \(b\) are any integers such that there is a divisor \(D\) on \(X\) with \(D\cdot af\) coprime to \(b\). The paper constructs an equivalence of categories between the derived categories of coherent sheaves on \(X\) and \(J_X (a,b)\). – As an application it is shown that if \(E\) is a stable vector bundle on \(X\) (with respect to a suitable polarization) whose rank \(r\) and fibre degree \(d=c_1 (E)\cdot f\) are coprime, then the component of the moduli space of stable sheaves on \(X\) containing \(E\) is birationally equivalent to a Hilbert scheme of points on \(J_X (a,b)\), where \(a\) and \(b\) are the unique pair of integers satisfying \(br-a d=1\) and \(0<a<r\).

14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
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