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)



The resurgence of ideals of points and the containment problem

Authors: Cristiano Bocci and Brian Harbourne
Journal: Proc. Amer. Math. Soc. 138 (2010), 1175-1190
MSC (2010): Primary 14C20, 13C05; Secondary 14N05, 14H20, 41A05
Published electronically: December 9, 2009
MathSciNet review: 2578512
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We relate properties of linear systems on $ X$ to the question of when $ I^r$ contains $ I^{(m)}$ in the case that $ I$ is the homogeneous ideal of a finite set of distinct points $ p_1,\ldots,p_n\in\mathbf{P}^2$, where $ X$ is the surface obtained by blowing up the points. We obtain complete answers for when $ I^r$ contains $ I^{(m)}$ when the points $ p_i$ lie on a smooth conic or when the points are general and $ n\le 9$.

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

  • [PSC] T. Bauer, S. Di Rocco, B. Harbourne, M. Kapustka, A. Knutsen, W. Syzdek, and T. Szemberg. A primer on Seshadri constants, to appear in the AMS Contemporary Mathematics series volume ``Interactions of Classical and Numerical Algebraic Geometry,'' proceedings of a conference in honor of A.J. Sommese, held at Notre Dame, May 22-24, 2008.
  • [BH] C. Bocci and B. Harbourne. Comparing powers and symbolic powers of ideals, Journal of Algebraic Geometry, to appear.
  • [C] G. Castelnuovo. Ricerche generali sopra i sistemi lineari di curve piane, Mem. Accad. Sci. Torino, II 42 (1891).
  • [CM] C. Ciliberto and R. Miranda. Nagata's conjecture for a square number of points, Ricerche di Matematica 55 (2006), 71-78. MR 2248163 (2007d:14098)
  • [ELS] L. Ein, R. Lazarsfeld and K. Smith. Uniform bounds and symbolic powers on smooth varieties, Invent. Math. 144 (2001), 241-252. MR 1826369 (2002b:13001)
  • [E] L. Evain. Computing limit linear series with infinitesimal methods, Ann. Inst. Fourier (Grenoble) 57 (2007), 1947-1974. MR 2377892 (2009e:14044)
  • [G] A. Gimigliano. On linear systems of plane curves, Thesis, Queen's University, Kingston, ON, Canada, 1987.
  • [GuH] E. Guardo and B. Harbourne. Resolutions of ideals of six fat points in $ \mathbf{P}^2$, J. Alg. 318 (2) (2007), 619-640. MR 2371962 (2009a:13019)
  • [H1] B. Harbourne. Birational models of rational surfaces, J. Alg. 190 (1997), 145-162. MR 1442149 (98a:14023)
  • [H2] B. Harbourne. Free resolutions of fat point ideals on $ \mathbf{P}^2$, J. Pure Appl. Alg. 125 (1998), 213-234. MR 1600024 (99d:13016)
  • [H3] B. Harbourne. An algorithm for fat points on $ \mathbf{P}^2$, Can. J. Math. 52 (2000), 123-140. MR 1745704 (2001g:13037)
  • [H4] B. Harbourne. The geometry of rational surfaces and Hilbert functions of points in the plane, Proceedings of the 1984 Vancouver Conference in Algebraic Geometry, CMS Conf. Proc., 6, Amer. Math. Soc., Providence, RI, 1986, pp. 95-111. MR 846019 (87k:14041)
  • [H5] B. Harbourne. Rational surfaces with $ K^2>0$, Proc. Amer. Math. Soc. 124 (1996), 727-733. MR 1307526 (96f:14045)
  • [H6] B. Harbourne. Complete linear systems on rational surfaces, Trans. Amer. Math. Soc. 289 (1985), 213-226. MR 779061 (86h:14030)
  • [H7] B. Harbourne. Global aspects of the geometry of surfaces, Ann. Univ. Paedagogicae Cracoviensis Studia Mathematica, to appear.
  • [Hr] R. Hartshorne. Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, New York, 1977. MR 0463157 (57:3116)
  • [Hi] A. Hirschowitz. Une conjecture pour la cohomologie des diviseurs sur les surfaces rationelles génériques, J. Reine Angew. Math. 397 (1989), 208-213. MR 993223 (90g:14021)
  • [Ho] M. Hochster. Criteria for equality of ordinary and symbolic powers of primes, Math. Z. 133 (1973), 53-65. MR 0323771 (48:2127)
  • [HoH] M. Hochster and C. Huneke. Comparison of symbolic and ordinary powers of ideals, Invent. Math. 147 (2002), no. 2, 349-369. MR 1881923 (2002m:13002)
  • [HH] S. Huckaba and C. Huneke. Powers of ideals having small analytic deviation, Amer. J. Math. 114 (1992), 367-403. MR 1156570 (93g:13002)
  • [LS] A. Li and I. Swanson. Symbolic powers of radical ideals, Rocky Mountain J. of Math. 36 (2006), 997-1009. MR 2254374 (2007k:13004)
  • [M] H. Matsumura. Commutative Algebra. W. A. Benjamin, New York, 1970. MR 0266911 (42:1813)
  • [Mo] S. Morey. Stability of associated primes and equality of ordinary and symbolic powers of ideals, Comm. Alg. 27(7) (1999), 3221-3231. MR 1695295 (2000e:13034)
  • [MNV] S. Morey, S. Noh and W. Vasconcelos. Symbolic powers, Serre conditions and Cohen-Macaulay Rees algebras. Manuscripta Math. 86 (1995), 113-124. MR 1314152 (96b:13002)
  • [R] J. Roé. Limit linear systems and applications, preprint (arXiv: math.AG/0602213.pdf)
  • [S] B. Segre. Alcune questioni su insiemi finiti di punti in geometria algebrica, Atti Convegno Intern. di Geom. Alg. di Torino (1961), Rattero, Turin, 1962, pp. 15-33. MR 0146714 (26:4234)
  • [T] H. Terakawa. The $ d$-very ampleness on a projective surface in characteristic $ p$, Pac. J. Math. 187 (1999), 187-199. MR 1674325 (99m:14014)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14C20, 13C05, 14N05, 14H20, 41A05

Retrieve articles in all journals with MSC (2010): 14C20, 13C05, 14N05, 14H20, 41A05

Additional Information

Cristiano Bocci
Affiliation: Dipartimento di Scienze Matematiche e Informatiche “R. Magari”, Università degli Studi di Siena, Pian dei mantellini, 44, 53100 Siena, Italy

Brian Harbourne
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588-0130

Keywords: Fat points, symbolic powers, normal generation, projective space
Received by editor(s): March 14, 2008
Received by editor(s) in revised form: May 26, 2009
Published electronically: December 9, 2009
Additional Notes: This research was partially supported by GNSAGA of INdAM (Italy) and by the NSA
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society