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)

 
 

 

Higher order embeddings of certain blow-ups of $ \mathbb{P}^2$


Authors: Cindy De Volder and Halszka Tutaj-Gasinska
Journal: Proc. Amer. Math. Soc. 137 (2009), 4089-4097
MSC (2000): Primary 14C20
DOI: https://doi.org/10.1090/S0002-9939-09-10037-0
Published electronically: July 10, 2009
MathSciNet review: 2538570
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X_n$ be the blow-up of the projective plane along $ n$ general points of a smooth cubic plane curve and let $ \mathcal{L}$ be the linear series of strict transforms of plane curves of degree $ d$ having multiplicity at least $ m_i$ at the $ i$-th blown-up point. We prove that if $ \mathcal{L}$ is $ k$-very ample, then $ \mathcal{L}$ is excellent and $ \mathcal{L}\cdot (-K_n) \geq k+2$. Then we give a numerical criterion for the $ k$-very ampleness of excellent classes with $ \mathcal{L} \cdot (-K_n) \geq k+2$, which in many cases is a necessary and sufficient condition.


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

  • 1. E. Ballico, M. Coppens, Very ample line bundles on blown-up projective varieties, Bull. Belg. Math. Soc. Simon Stevin, 4 (1997), 437-447. MR 1476035 (98k:14019)
  • 2. E. Ballico, A.J. Sommese, Projective surfaces with $ k$-very ample line bundles of degree $ \leq 4k+4$, Nagoya Math. J., 136 (1994), 57-79. MR 1309381 (96d:14005)
  • 3. M. Beltrametti, P. Francia, A.J. Sommese, On Reider's method and higher order embeddings, Duke Math. J., 58 (1989), 425-439. MR 1016428 (90h:14021)
  • 4. M. Beltrametti, A.J. Sommese, On $ k$-spannedness for projective surfaces, Algebraic geometry (L'Aquila, 1988), Lecture Notes in Math., vol. 1417, Springer, Berlin, 1990, pp. 24-51. MR 1040549 (91g:14029)
  • 5. M. Beltrametti, A.J. Sommese, Zero cycles and $ k$th order embeddings of smooth projective surfaces, in ``Problems in the theory of surfaces and their classification'' (Cortona, 1988), Sympos. Math., XXXII, Academic Press, London, 1991, pp. 33-48. MR 1273371 (95d:14005)
  • 6. M. Beltrametti, A.J. Sommese, On $ k$-jet ampleness, in ``Complex analysis and geometry'', ed. by Ancona and Silva, Plenum Press, New York, 1993, pp. 355-376. MR 1211891 (94g:14006)
  • 7. M. Beltrametti, A.J. Sommese, Notes on embeddings of blowups, J. Algebra, 186 (1996), 861-871. MR 1424597 (97m:14004)
  • 8. S. Chauvin, C. De Volder, Some very ample and base point free linear systems on generic rational surfaces, Math. Nachr., 245 (2002), 45-66. MR 1936343 (2004c:14007)
  • 9. C. De Volder, A. Laface, A note on the very ampleness of complete linear systems on blowings-up of $ \mathbb{P}\sp 3$, in ``Projective varieties with unexpected properties'', Walter de Gruyter, Berlin, 2005, pp. 231-236. MR 2202255 (2006i:14004)
  • 10. B. Harbourne, Complete linear systems on rational surfaces, Trans. Amer. Math. Soc., 289 (1985), 213-226. MR 779061 (86h:14030)
  • 11. B. Harbourne, Very ample divisors on rational surfaces, Math. Ann., 272 (1985), 139-153. MR 794097 (86k:14026)
  • 12. E. Looijenga, Rational surfaces with an anticanonical cycle, Ann. of Math. (2), 114 (1981), 267-322. MR 632841 (83j:14030)
  • 13. Y.I. Manin, Cubic forms, North-Holland Mathematical Library, vol. 4, North-Holland Publishing Co., Amsterdam, 1986. MR 833513 (87d:11037)
  • 14. T. Szemberg, H. Tutaj-Gasińska, General blow-ups of the projective plane, Proc. Amer. Math. Soc., 130 (2002), 2515-2524. MR 1900857 (2003b:14013)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 14C20

Retrieve articles in all journals with MSC (2000): 14C20


Additional Information

Cindy De Volder
Affiliation: Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281, S22, B-9000 Ghent, Belgium
Email: cindy.devolder@ugent.be

Halszka Tutaj-Gasinska
Affiliation: Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, PL-30348 Kraków, Poland – and – Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, PL-00956 Warszawa, Poland
Email: htutaj@im.uj.edu.pl

DOI: https://doi.org/10.1090/S0002-9939-09-10037-0
Keywords: $k$-very ample, anticanonical rational surface
Received by editor(s): May 17, 2008
Received by editor(s) in revised form: January 24, 2009, and April 30, 2009
Published electronically: July 10, 2009
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