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)

 
 

 

Singular hypersurfaces possessing infinitely many star points


Authors: Filip Cools and Marc Coppens
Journal: Proc. Amer. Math. Soc. 139 (2011), 3413-3422
MSC (2010): Primary 14J70, 14N15, 14N20
DOI: https://doi.org/10.1090/S0002-9939-2011-10760-3
Published electronically: March 3, 2011
MathSciNet review: 2813373
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that a component $ \Lambda$ of the closure of the set of star points on a hypersurface of degree $ d\geq 3$ in $ \mathbb{P}^N$ is linear. Afterwards, we focus on the case where $ \Lambda$ is of maximal dimension and the case where $ X$ is a surface.


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

  • 1. E. Ciani, Sopra le superficie cubiche dotate di infiniti punti di Eckardt, Periodici di Mat. 20 (1940), 240-245. MR 0004489 (3:14i)
  • 2. F. Cools, M. Coppens, Star points on smooth hypersurfaces, J. Algebra 323 (2010), 261-286. MR 2564838
  • 3. F.E. Eckardt, Ueber diejenigen Flächen dritter Grades, auf denen sich drei geraden Linien in einem Punkte schneiden, Math. Ann. 10 (1876), 227-272. MR 1509887
  • 4. G. Fischer, J. Piontkowski, Ruled Varieties, Advanced Lectures in Mathematics, Vieweg, 2001. MR 1876644 (2003f:14044)
  • 5. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, 1977. MR 0463157 (57:3116)
  • 6. S. Kleiman, Bertini and his two fundamental theorems, Rend. Circ. Mat. Palermo 55 (1998), 9-37. MR 1661859 (99m:14001)
  • 7. E. Looijenga, The complement of the bifurcation variety of a simple singularity, Invent. Math. 23 (1974), 105-116. MR 0422675 (54:10661)
  • 8. O.V. Lyashko, Decomposition of simple singularities of functions, Funkts. Anal. Prilozh. 10 (1976), 49-56. MR 0414929 (54:3021)
  • 9. T.C. Nguyen, Star points on cubic surfaces, Ph.D. Thesis, RU Utrecht, 2000.
  • 10. T.C. Nguyen, Non-singular cubic surfaces with star points, Vietnam J. Math. 29 (2001), 287-292. MR 1933910 (2003i:14068)
  • 11. T.C. Nguyen, On boundaries of moduli spaces of non-singular cubic surfaces with star points, Kodai Math. J. 27 (2004), 57-73. MR 2042791 (2004m:14072)
  • 12. T.C. Nguyen, On semi-stable, singular cubic surfaces, Séminaires et Congrès 10 (2005), 373-389. MR 2145966 (2006c:14060)
  • 13. F. Zak, Tangents and secants of algebraic varieties, Transl. Math. Monogr., 127, Amer. Math. Soc., 1993. MR 1234494 (94i:14053)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14J70, 14N15, 14N20

Retrieve articles in all journals with MSC (2010): 14J70, 14N15, 14N20


Additional Information

Filip Cools
Affiliation: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium
Email: Filip.Cools@wis.kuleuven.be

Marc Coppens
Affiliation: Departement Industriel Ingenieur en Biotechniek, Katholieke Hogeschool Kempen, Kleinhoefstraat 4, B-2440 Geel, Belgium – and – Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium
Email: Marc.Coppens@khk.be

DOI: https://doi.org/10.1090/S0002-9939-2011-10760-3
Received by editor(s): January 18, 2010
Received by editor(s) in revised form: August 19, 2010, and August 26, 2010
Published electronically: March 3, 2011
Communicated by: Lev Borisov
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society