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On the essential dimension of coherent sheaves

  • Indranil Biswas EMAIL logo , Ajneet Dhillon and Norbert Hoffmann

Abstract

We characterize all fields of definition for a given coherent sheaf over a projective scheme in terms of projective modules over a finite-dimensional endomorphism algebra. This yields general results on the essential dimension of such sheaves. Applying them to vector bundles over a smooth projective curve C, we obtain an upper bound for the essential dimension of their moduli stack. The upper bound is sharp if the conjecture of Colliot-Thélène, Karpenko and Merkurjev holds. We find that the genericity property proved for Deligne–Mumford stacks by Brosnan, Reichstein and Vistoli still holds for this Artin stack, unless the curve C is elliptic.

Award Identifier / Grant number: SFB 647 Space–Time–Matter

Funding statement: I.B. was supported by the J. C. Bose Fellowship. A.D. was partially supported by NSERC. N.H. was partially supported by SFB 647 “Space–Time–Matter” of the DFG (German Research Foundation).

Acknowledgements

We are grateful to the referee for helpful suggestions. N.H. thanks TIFR Bombay for hospitality, and Bernd Kreussler for a useful discussion on bundles over elliptic curves.

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Received: 2014-12-19
Published Online: 2015-7-10
Published in Print: 2018-2-1

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