Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Dominance of a rational map to the Coble quartic

Author: Sukmoon Huh
Journal: Proc. Amer. Math. Soc. 138 (2010), 777-786
MSC (2000): Primary 14D20; Secondary 14M15
Published electronically: October 20, 2009
MathSciNet review: 2566543
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show the dominance of the restriction map from a moduli space of stable sheaves on the projective plane to the Coble sixfold quartic. With the dominance and the interpretation of a stable sheaf on the plane in terms of hyperplane arrangements, we expect these tools to reveal the geometry of the Coble quartic.

References [Enhancements On Off] (What's this?)

  • 1. E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves, Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932 (86h:14019)
  • 2. W. Barth, Moduli of vector bundles on the projective plane, Invent. Math. 42 (1977), 63-91. MR 0460330 (57:324)
  • 3. Arthur B. Coble, Algebraic geometry and theta functions, revised printing. American Mathematical Society Colloquium Publication, vol. X, American Mathematical Society, Providence, RI, 1961. MR 0123958 (23:A1279)
  • 4. I. Dolgachev and M. Kapranov, Arrangements of hyperplanes and vector bundles on $ \mathbf P\sp n$, Duke Math. J. 71 (1993), no. 3, 633-664. MR 1240599 (95e:14029)
  • 5. G. Elencwajg, Les fibrés uniformes de rang $ 3$ sur $ {\bf P}\sb{2}({\bf C})$ sont homogènes, Math. Ann. 231 (1977/78), no. 3, 217-227. MR 0481133 (58:1278)
  • 6. Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York, 1977. MR 0463157 (57:3116)
  • 7. G. Hein, The tangent bundle of $ {\bf P}\sp 2$ restricted to plane curves, Complex analysis and geometry (Trento, 1995), Pitman Res. Notes Math. Ser., vol. 366, Longman, Harlow, 1997, pp. 137-140. MR 1477446 (98j:14047)
  • 8. S. Huh, Moduli spaces of stable sheaves on the projective plane and on the plane quartic curve, Ph.D. thesis, University of Michigan, May 2007.
  • 9. Klaus Hulek, Stable rank-$ 2$ vector bundles on $ {\bf P}\sb{2}$ with $ c\sb{1}$ odd, Math. Ann. 242 (1979), no. 3, 241-266. MR 545217 (80m:14011)
  • 10. Young-Hoon Kiem, The stringy $ E$-function of the moduli space of rank $ 2$ bundles over a Riemann surface of genus $ 3$, Trans. Amer. Math. Soc. 355 (2003), no. 5, 1843-1856 (electronic). MR 1953528 (2003j:14045)
  • 11. Herbert Lange, Higher secant varieties of curves and the theorem of Nagata on ruled surfaces, Manuscripta Math. 47 (1984), no. 1-3, 263-269. MR 744323 (85f:14043)
  • 12. Shigeru Mukai, An introduction to invariants and moduli, Cambridge Studies in Advanced Mathematics, vol. 81, Cambridge University Press, Cambridge, 2003; translated from the 1998 and 2000 Japanese editions by W. M. Oxbury. MR 2004218 (2004g:14002)
  • 13. M. S. Narasimhan and S. Ramanan, $ 2\theta$-linear systems on abelian varieties, Vector bundles on algebraic varieties (Bombay, 1984), Tata Inst. Fund. Res. Stud. Math., vol. 11, Tata Inst. Fund. Res., Bombay, 1987, pp. 415-427. MR 893605 (88j:14014)
  • 14. W. M. Oxbury, C. Pauly, and E. Previato, Subvarieties of $ {SU}\sb C(2)$ and $ 2\theta$-divisors in the Jacobian, Trans. Amer. Math. Soc. 350 (1998), no. 9, 3587-3614. MR 1467474 (98m:14034)
  • 15. J. G. Semple and L. Roth, Introduction to algebraic geometry, Oxford, at the Clarendon Press, 1949. MR 0034048 (11:535d)
  • 16. F. L. Zak, Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, vol. 127, American Mathematical Society, Providence, RI, 1993; translated from the Russian manuscript by the author. MR 1234494 (94i:14053)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 14D20, 14M15

Retrieve articles in all journals with MSC (2000): 14D20, 14M15

Additional Information

Sukmoon Huh
Affiliation: Korea Institute for Advanced Study, Hoegiro 87, Dongdaemun-gu, Seoul 130-722, Republic of Korea

Received by editor(s): January 15, 2008
Received by editor(s) in revised form: April 26, 2009
Published electronically: October 20, 2009
Additional Notes: This article is part of the revised version of the author’s thesis at the University of Michigan. The author would like to express his deepest gratitude to his advisor, Professor Igor Dolgachev. The author is also grateful to the referee for many suggestions.
Communicated by: Ted Chinburg
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society