# On the essential dimension of coherent sheaves

• Indranil Biswas , 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.

## References

[1] M. F. Atiyah, On the Krull-Schmidt theorem with application to sheaves, Bull. Soc. Math. France 84 (1956), 307–317. 10.24033/bsmf.1475Search in Google Scholar

[2] V. Balaji, I. Biswas, O. Gabber and D. S. Nagaraj, Brauer obstruction for a universal vector bundle, C.R. Math. Acad. Sci. Paris 345 (2007), 265–268. 10.1016/j.crma.2007.07.011Search in Google Scholar

[3] I. Biswas, A. Dhillon and N. Lemire, The essential dimension of stacks of parabolic bundles over curves, J. K-Theory 10 (2012), 455–488. 10.1017/is011008026jkt192Search in Google Scholar

[4] I. Biswas and N. Hoffmann, Some moduli stacks of symplectic bundles on a curve are rational, Adv. Math. 219 (2008), 1150–1176. 10.1016/j.aim.2008.06.001Search in Google Scholar

[5] S. Bosch, W. Lütkebohmert and M. Raynaud, Néron models, Ergeb. Math. Grenzgeb. (3) 21, Springer, Berlin 1990. 10.1007/978-3-642-51438-8Search in Google Scholar

[6] P. Brosnan, Z. Reichstein and A. Vistoli, Essential dimension of moduli of curves and other algebraic stacks. With an appendix by N. Fakhruddin, J. Eur. Math. Soc. 13 (2011), 1079–1112. 10.4171/JEMS/276Search in Google Scholar

[7] J. Buhler and Z. Reichstein, On the essential dimension of a finite group, Compos. Math. 106 (1997), 159–179. 10.1023/A:1000144403695Search in Google Scholar

[8] J.-L. Colliot-Thélène, N. A. Karpenko and A. S. Merkurjev, Rational surfaces and the canonical dimension of the group PGL6, Algebra i Analiz 19 (2007), 159–178. Search in Google Scholar

[9] A. Dhillon and N. Lemire, Upper bounds for the essential dimension of the moduli stack of SLn-bundles over a curve, Transform. Groups 14 (2009), 747–770. 10.1007/s00031-009-9069-6Search in Google Scholar

[10] J.-M. Drezet and M. S. Narasimhan, Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. Math. 97 (1989), 53–94. 10.1007/BF01850655Search in Google Scholar

[11] T. L. Gómez, Algebraic stacks, Proc. Indian Acad. Sci. Math. Sci. 111 (2001), 1–31. 10.1007/BF02829538Search in Google Scholar

[12] N. Hoffmann, Rationality and Poincaré families for vector bundles with extra structure on a curve, Int. Math. Res. Not. (2007), Article ID rnm010. 10.1093/imrn/rnm010Search in Google Scholar

[13] N. Hoffmann, Moduli stacks of vector bundles on curves and the King–Schofield rationality proof, Cohomological and geometric approaches to rationality problems, Progr. Math. 282, Birkhäuser, Boston (2010), 133–148. 10.1007/978-0-8176-4934-0_5Search in Google Scholar

[14] N. Hoffmann, The moduli stack of vector bundles on a curve, Teichmüller theory and moduli problems, Ramanujan Math. Soc. Lect. Notes Ser. 10, (()2010), 387–394. Search in Google Scholar

[15] D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, 2nd ed., Cambridge University Press, Cambridge 2010. 10.1017/CBO9780511711985Search in Google Scholar

[16] I. M. Isaacs, Algebra: A graduate course, Brooks/Cole Publishing Co., Pacific Grove 1993. Search in Google Scholar

[17] N. A. Karpenko, Upper motives of algebraic groups and incompressibility of Severi–Brauer varieties, J. reine angew. Math. 677 (2013), 179–198. 10.1515/crelle.2012.011Search in Google Scholar

[18] G. Laumon, Un analogue global du cône nilpotent, Duke Math. J. 57 (1988), 647–671. 10.1215/S0012-7094-88-05729-8Search in Google Scholar

[19] G. Laumon and L. Moret-Bailly, Champs algébriques, Ergeb. Math. Grenzgeb. (3) 39, Springer, Berlin 2000. 10.1007/978-3-540-24899-6Search in Google Scholar

[20] J. Le Potier, Lectures on vector bundles, Cambridge University Press, Cambridge 1997. Search in Google Scholar

[21] A. S. Merkurjev, Essential dimension: A survey, Transform. Groups 18 (2013), 415–481. 10.1007/s00031-013-9216-ySearch in Google Scholar

[22] A. Polishchuk, Abelian varieties, theta functions and the Fourier transform, Cambridge University Press, Cambridge 2003. 10.1017/CBO9780511546532Search in Google Scholar

[23] Z. Reichstein, Essential dimension, Proceedings of the international congress of mathematicians. Vol. II (ICM 2010), Hindustan Book Agency, New Delhi (2011), 162–188. 10.1142/9789814324359_0045Search in Google Scholar

[24] Z. Reichstein and A. Vistoli, A genericity theorem for algebraic stacks and essential dimension of hypersurfaces, J. Eur. Math. Soc. 15 (2013), 1999–2026. 10.4171/JEMS/411Search in Google Scholar