Comparing powers and symbolic powers of ideals

Bocci, Cristiano; Harbourne, Brian

doi:10.1090/S1056-3911-09-00530-X

Authors: Cristiano Bocci and Brian Harbourne
Journal: J. Algebraic Geom. 19 (2010), 399-417
DOI: https://doi.org/10.1090/S1056-3911-09-00530-X
Published electronically: August 17, 2009
MathSciNet review: 2629595
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Abstract | References | Additional Information

Abstract: We develop tools to study the problem of containment of symbolic powers $I^{(m)}$ in powers $I^r$ for a homogeneous ideal $I$ in a polynomial ring $k[\textbf {P}^N]$ in $N+1$ variables over an arbitrary algebraically closed field $k$. We obtain results on the structure of the set of pairs $(r,m)$ such that $I^{(m)}\subseteq I^r$. As corollaries, we show that $I^2$ contains $I^{(3)}$ whenever $S$ is a finite generic set of points in $\textbf {P}^2$ (thereby giving a partial answer to a question of Huneke), and we show that the containment theorems of Ein–Lazarsfeld–Smith [Invent. Math. 144 (2001), pp. 241–252] and Hochster–Huneke [Invent. Math. 147 (2002), pp. 349–369] are optimal for every fixed dimension and codimension.

References

Alessandro Arsie and Jon Eivind Vatne, A note on symbolic and ordinary powers of homogeneous ideals, Ann. Univ. Ferrara Sez. VII (N.S.) 49 (2003), 19–30 (English, with English and Italian summaries). MR 2164987
M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
Paul Biran, Constructing new ample divisors out of old ones, Duke Math. J. 98 (1999), no. 1, 113–135. MR 1687571, DOI https://doi.org/10.1215/S0012-7094-99-09803-4

C. Bocci and B. Harbourne. The resurgence of ideals of points and the containment problem, to appear, Proc. Amer. Math. Soc.

M. Chardin. Regularity of ideals and their powers, Prépublication 364, Institut de Mathematiques de Jussieu, 2004.

Marc Chardin, Some results and questions on Castelnuovo-Mumford regularity, Syzygies and Hilbert functions, Lect. Notes Pure Appl. Math., vol. 254, Chapman & Hall/CRC, Boca Raton, FL, 2007, pp. 1–40. MR 2309925, DOI https://doi.org/10.1201/9781420050912.ch1

Steven Dale Cutkosky, Huy Tài Hà, Hema Srinivasan, and Emanoil Theodorescu, Asymptotic behavior of the length of local cohomology, Canad. J. Math. 57 (2005), no. 6, 1178–1192. MR 2178557, DOI https://doi.org/10.4153/CJM-2005-046-4

Jean-Pierre Demailly, Singular Hermitian metrics on positive line bundles, Complex algebraic varieties (Bayreuth, 1990) Lecture Notes in Math., vol. 1507, Springer, Berlin, 1992, pp. 87–104. MR 1178721, DOI https://doi.org/10.1007/BFb0094512

Lawrence Ein, Robert Lazarsfeld, and Karen E. Smith, Uniform bounds and symbolic powers on smooth varieties, Invent. Math. 144 (2001), no. 2, 241–252. MR 1826369, DOI https://doi.org/10.1007/s002220100121

David Eisenbud, Craig Huneke, and Bernd Ulrich, The regularity of Tor and graded Betti numbers, Amer. J. Math. 128 (2006), no. 3, 573–605. MR 2230917

Anthony V. Geramita, Alessandro Gimigliano, and Yves Pitteloud, Graded Betti numbers of some embedded rational $n$-folds, Math. Ann. 301 (1995), no. 2, 363–380. MR 1314592, DOI https://doi.org/10.1007/BF01446634

A. V. Geramita, B. Harbourne and J. Migliore. Classifying Hilbert functions of fat point subschemes in $\textbf {P}^2$, Collect. Math. 60, 2 (2009), 159–192.

A. V. Geramita, J. Migliore, and L. Sabourin, On the first infinitesimal neighborhood of a linear configuration of points in $\Bbb P^2$, J. Algebra 298 (2006), no. 2, 563–611. MR 2217628, DOI https://doi.org/10.1016/j.jalgebra.2006.01.035

Elena Guardo and Brian Harbourne, Resolutions of ideals of any six fat points in ${\bf P}^2$, J. Algebra 318 (2007), no. 2, 619–640. MR 2371962, DOI https://doi.org/10.1016/j.jalgebra.2007.09.018

Brian Harbourne, Seshadri constants and very ample divisors on algebraic surfaces, J. Reine Angew. Math. 559 (2003), 115–122. MR 1989646, DOI https://doi.org/10.1515/crll.2003.044

Brian Harbourne, Complete linear systems on rational surfaces, Trans. Amer. Math. Soc. 289 (1985), no. 1, 213–226. MR 779061, DOI https://doi.org/10.1090/S0002-9947-1985-0779061-2

Brian Harbourne, An algorithm for fat points on $\mathbf P^2$, Canad. J. Math. 52 (2000), no. 1, 123–140. MR 1745704, DOI https://doi.org/10.4153/CJM-2000-006-6

Brian Harbourne, On Nagata’s conjecture, J. Algebra 236 (2001), no. 2, 692–702. MR 1813496, DOI https://doi.org/10.1006/jabr.2000.8515

Brian Harbourne and Joaquim Roé, Discrete behavior of Seshadri constants on surfaces, J. Pure Appl. Algebra 212 (2008), no. 3, 616–627. MR 2365336, DOI https://doi.org/10.1016/j.jpaa.2007.06.018

B. Harbourne and J. Roé. Extendible Estimates of multipoint Seshadri Constants, preprint, math.AG/0309064, 2003.

B. Harbourne and J. Roé. Computing multi-point Seshadri constants on $\textbf {P}^2$, to appear, Bulletin of the Belgian Mathematical Society - Simon Stevin.

Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157

André Hirschowitz, La méthode d’Horace pour l’interpolation à plusieurs variables, Manuscripta Math. 50 (1985), 337–388 (French, with English summary). MR 784148, DOI https://doi.org/10.1007/BF01168836

Melvin Hochster, Criteria for equality of ordinary and symbolic powers of primes, Math. Z. 133 (1973), 53–65. MR 323771, DOI https://doi.org/10.1007/BF01226242

Melvin Hochster and Craig Huneke, Comparison of symbolic and ordinary powers of ideals, Invent. Math. 147 (2002), no. 2, 349–369. MR 1881923, DOI https://doi.org/10.1007/s002220100176

Melvin Hochster and Craig Huneke, Fine behavior of symbolic powers of ideals, Illinois J. Math. 51 (2007), no. 1, 171–183. MR 2346193

Vijay Kodiyalam, Asymptotic behaviour of Castelnuovo-Mumford regularity, Proc. Amer. Math. Soc. 128 (2000), no. 2, 407–411. MR 1621961, DOI https://doi.org/10.1090/S0002-9939-99-05020-0

Aihua Li and Irena Swanson, Symbolic powers of radical ideals, Rocky Mountain J. Math. 36 (2006), no. 3, 997–1009. MR 2254374, DOI https://doi.org/10.1216/rmjm/1181069441

Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911

Thierry Mignon, Systèmes de courbes planes à singularités imposées: le cas des multiplicités inférieures ou égales à quatre, J. Pure Appl. Algebra 151 (2000), no. 2, 173–195 (French, with English summary). MR 1775572, DOI https://doi.org/10.1016/S0022-4049%2899%2900054-7

Masayoshi Nagata, On rational surfaces. II, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 33 (1960/61), 271–293. MR 126444, DOI https://doi.org/10.1215/kjm/1250775912

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.

Joaquim Roé, A relation between one-point and multi-point Seshadri constants, J. Algebra 274 (2004), no. 2, 643–651. MR 2043369, DOI https://doi.org/10.1016/j.jalgebra.2003.10.009

Irena Swanson, Linear equivalence of ideal topologies, Math. Z. 234 (2000), no. 4, 755–775. MR 1778408, DOI https://doi.org/10.1007/s002090050007

Tomasz Szemberg and Halszka Tutaj-Gasińska, General blow-ups of the projective plane, Proc. Amer. Math. Soc. 130 (2002), no. 9, 2515–2524. MR 1900857, DOI https://doi.org/10.1090/S0002-9939-02-06488-2

S. Takagi and K. Yoshida. Generalized test ideals and symbolic powers, preprint, 2007, math.AC/0701929.

Zachariah C. Teitler, On the intersection of the curves through a set of points in $\Bbb P^2$, J. Pure Appl. Algebra 209 (2007), no. 2, 571–581. MR 2293328, DOI https://doi.org/10.1016/j.jpaa.2006.07.007

Halszka Tutaj-Gasińska, A bound for Seshadri constants on ${\Bbb P}^2$, Math. Nachr. 257 (2003), 108–116. MR 1992815, DOI https://doi.org/10.1002/mana.200310082

Geng Xu, Ample line bundles on smooth surfaces, J. Reine Angew. Math. 469 (1995), 199–209. MR 1363830, DOI https://doi.org/10.1515/crll.1995.469.199

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
Email: bocci24@unisi.it

Brian Harbourne
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588-0130
MR Author ID: 217048
Email: bharbour@math.unl.edu

Received by editor(s): February 12, 2008
Published electronically: August 17, 2009
Additional Notes: This reseach was partially supported by Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni of Istituto Nazionale di Alta Matematica (Italy) and by the National Security Agency