# Journal of the Ramanujan Mathematical Society

Volume 32, Issue 1, March 2017 pp. 43–50.

Fourier-Mukai transform of vector bundles on surfaces to Hilbert scheme

**Authors**:
Indranil Biswas and D. S. Nagaraj

**Author institution:**School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India

**Summary: **
Let S be an irreducible smooth projective surface defined over
an algebraically closed field k. For a positive integer d, let
Hilb d(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) →
Hilb d(S) be its Fourier--Mukai transform constructed using the
structure sheaf of the universal subscheme of S ×
Hilb d(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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