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On the essential dimension of coherent sheaves

Research paper by Indranil Biswas, Ajneet Dhillon, Norbert Hoffmann

Indexed on: 02 Dec '14Published on: 02 Dec '14Published in: Mathematics - Algebraic Geometry



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-Th\'el\`ene, 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.