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Fourier-Mukai Transform of Vector Bundles on Surfaces to Hilbert Scheme
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Let S be an irreducible smooth projective surface defined over an algebraically closed field k. For a positive integer d, let Hilbd (S) be the Hilbert scheme parametrizing the zero-dimensional subschemes of S of length d. For a vector bundle E on S, let H(E) → Hilbd (S) be its Fourier–Mukai transform constructed using the structure sheaf of the universal subscheme of S × Hilbd (S) as the kernel. We prove that two vector bundles E and F on S are isomorphic if the vector bundles H(E) and H(F) are isomorphic.
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