Abstract
Let X be an irreducible smooth projective surface over \({{\mathbb{C}}}\) and Hilbd(X) the Hilbert scheme parametrizing the zero-dimensional subschemes of X of length d. Given a vector bundle E on X, there is a naturally associated vector bundle \({{\mathcal{F}}_d(E)}\) over Hilbd(X). If E and V are semistable vector bundles on X such that \({{\mathcal{F}}_d(E)}\) and \({{\mathcal{F}}_d(V)}\) are isomorphic, we prove that E is isomorphic to V. A key input in the proof is provided by Biswas and Nagaraj (see [1]).
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The first-named author acknowledges the support of the J. C. Bose Fellowship.
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Biswas, I., Nagaraj, D.S. On vector bundles over surfaces and Hilbert schemes. Arch. Math. 101, 513–517 (2013). https://doi.org/10.1007/s00013-013-0589-x
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DOI: https://doi.org/10.1007/s00013-013-0589-x