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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 $$0<a<r$$.

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