No Access

COHERENT SYSTEMS AND MODULAR SUBAVRIETIES OF

MICHELE BOLOGNESI

IRMAR, Université de Rennes 1, 263 Av. du Général Leclerc, 35042 Rennes, France

Search for more papers by this author

and

SONIA BRIVIO

Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy

Search for more papers by this author

https://doi.org/10.1142/S0129167X12500371Cited by:11 (Source: Crossref)

Abstract

Let C be an algebraic smooth complex curve of genus g > 1. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on C and the comparison of different type of notions of stability arising in moduli theory. Notably we show that in certain cases these moduli spaces are birationally equivalent to fibrations over simple projective varieties, whose fibers are GIT quotients (ℙ^r-1)^rg//PGL(r), where r is the rank of the considered vector bundles. This allows us to compare different definitions of (semi-)stability (slope stability, α-stability, GIT stability) for vector bundles, coherent systems and point sets, and derive relations between them. In certain cases of vector bundles of low rank when C has small genus, our construction produces families of classical modular varieties contained in the Coble hypersurfaces.

Keywords:

AMSC: 14H60, 14H10

References

A. Alzati and M. Bolognesi, A structure theorem for $\mathcal{SU}_C(2)$ and the moduli of pointed genus zero curves, preprint (2009), hyyp://arxiv.org/0903.5515, 1–20 . Google Scholar
C. Anghel, Serdica Math. J. 30(3), 103 (2004). Google Scholar
E. Arbarello et al. , Geometry of Algebraic Curves , Grundlehren der Mathematischen Wissenschaften 267 ( Springer-Verlag , New York , 1985 ) . Crossref, Google Scholar
A. Beauville, Amer. J. Math. 128(3), 607 (2006), DOI: 10.1353/ajm.2006.0018. Crossref, Web of Science, Google Scholar
A. Beauville, M. S. Narasimhan and S. Ramanan, J. Reine Angew. Math. 398, 169 (1989). Web of Science, Google Scholar
A. Bertram, J. Differential Geom. 35(2), 429 (1992). Crossref, Web of Science, Google Scholar
M. Bolognesi, Math. Z. 261(1), 149 (2009), DOI: 10.1007/s00209-008-0319-4. Crossref, Web of Science, Google Scholar
S. B. Bradlowet al., Comm. Algebra 37(8), 2649 (2009), DOI: 10.1080/00927870902747464. Crossref, Web of Science, Google Scholar
S. B. Bradlowet al., Internat. J. Math. 14(7), 683 (2003), DOI: 10.1142/S0129167X03002009. Link, Web of Science, Google Scholar
S. Brivio and A. Verra, Duke Math. J. 82, 503 (1996), DOI: 10.1215/S0012-7094-96-08222-8. Crossref, Web of Science, Google Scholar
S. Brivio and A. Verra, Algebraic Cycles and Motives, London Mathematical Society Lecture Note Series 344 (Cambridge University Press, Cambridge, 2007) pp. 73–93. Crossref, Google Scholar
S. Brivio and A. Verra, Plucker forms and the theta map, to appear in Amer. J. Math . Google Scholar
A. B. Coble , Algebraic Geometry and Theta Functions , Colloquium Publications 10 ( American Mathematical Society , Providence, RI , 1982 ) . Google Scholar
I. Dolgachev, Abstract configurations in algebraic geometry, The Fano Conference (Universitá di Torino, Turin, 2004) pp. 423–462. Google Scholar
I. Dolgachev and D. Ortland , Point Sets in the Projective Spaces and Theta Functions , Asterisque 165 ( Société Mathématique de France , 1988 ) . Google Scholar
J.-M. Drezet and M. S. Narasimhan, Invent. Math. 97(1), 53 (1989), DOI: 10.1007/BF01850655. Crossref, Web of Science, Google Scholar
D. Eisenbud and S. Popescu, J. Algebra 230(1), 127 (2000), DOI: 10.1006/jabr.1999.7940. Crossref, Web of Science, Google Scholar
B. Hunt , The Geometry of Some Special Arithmetic Quotients , Lecture Notes in Mathematics 1637 ( Springer , Berlin , 1996 ) . Crossref, Google Scholar
B. Hunt, Experiment. Math. 9(4), 613 (2000), DOI: 10.1080/10586458.2000.10504664. Crossref, Web of Science, Google Scholar
J.-I. Igusa, Amer. J. Math. 86, 219 (1964), DOI: 10.2307/2373041. Crossref, Web of Science, Google Scholar
E. Izadi and L. Van Geemen, J. Algebr. Geom. 10(1), 133 (2001). Web of Science, Google Scholar
A. D. King and P. E. Newstead, Internat. J. Math. 6(5), 733 (1995). Link, Web of Science, Google Scholar
A. King and A. Schofield, Indag. Math. (N.S.) 10(4), 519 (1999), DOI: 10.1016/S0019-3577(00)87905-7. Crossref, Web of Science, Google Scholar
K. Koike, Arch. Math. (Basel) 81(2), 155 (2003), DOI: 10.1007/s00013-003-4697-x. Crossref, Web of Science, Google Scholar
Y. Laszlo, Comment. Math. Helv. 71(3), 373 (1996), DOI: 10.1007/BF02566426. Crossref, Web of Science, Google Scholar
D. Mumford and P. Newstead, Amer. J. Math. 90, 1200 (1968), DOI: 10.2307/2373296. Crossref, Web of Science, Google Scholar
M. S. Narasimhan and S. Ramanan, Ann. of Math. (2) 89, 14 (1969), DOI: 10.2307/1970807. Crossref, Web of Science, Google Scholar
M. S. Narasimhan and S. Ramanan, Algebraic Geometry (Oxford University Press, London, 1969) pp. 335–346. Google Scholar
Q. M. Nguyen, Internat. J. Math. 18(5), 535 (2007). Link, Web of Science, Google Scholar
Q. M. Nguyen and S. Rams, Le Matematiche (Catania) 58(2), 257 (2005). Google Scholar
A. Ortega, J. Algebr. Geom. 14(2), 327 (2005), DOI: 10.1090/S1056-3911-04-00387-X. Crossref, Web of Science, Google Scholar
W. Oxbury and C. Pauly, Internat. J. Math. 7(3), 393 (1996). Link, Web of Science, Google Scholar
W. Oxbury and C. Pauly, Math. Proc. Cambridge Philos. Soc. 125(2), 295 (1999). Crossref, Web of Science, Google Scholar
C. Pauly, Michigan Math. J. 50(3), 551 (2002). Web of Science, Google Scholar
C. Pauly, Rank four vector bundles without theta divisor over a curve of genus two, to appear in Adv. Geom . Google Scholar
M. Raynaud, Bull. Soc. Math. France 110(1), 103 (1982). Web of Science, Google Scholar
A. Vistoli, Fundamental Algebraic Geometry, Mathematical Surveys Monographs 123 (American Mathematical Society, Providence, RI, 2005) pp. 1–104. Google Scholar