Abstract
We prove global effective versions of the Briançon–Skoda–Huneke theorem. Our results extend, to singular varieties, a result of Hickel on the membership problem in polynomial ideals in \({\mathbb C}^n\), and a related theorem of Ein and Lazarsfeld for smooth projective varieties. The proofs rely on known geometric estimates and new results on multivariable residue calculus.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
In Kollár’s theorem \(c_m=1\) even for \(m>n\) (unless \(d=2\), see also [30]) and this estimate is optimal.
In Kollár’s and Jelonek’s theorems, as well as in [21], there are more precise results that take into account different degree bounds \(d_j\) of \(F_j\), but for simplicity, in this paper we always keep all \(d_j=d\).
Often in the literature \(\mu +\mu _0\) is replaced by a constant independent of the number of generators \(m\).
The definition is the same when \(X\) is singular.
The sets \(Z_k^\text {\tiny {bef}}\) are the zero varieties of certain Fitting ideals associated with a free resolution of \({\mathcal O}^X/{\mathcal J}\); the importance of these sets (ideals) was pointed out by Buchsbaum and Eisenbud in the 70’s. We have not seen any notion for these sets in the literature, and “Buchsbaum–Eisenbud varieties” is already occupied for another purpose, so we stick to BEF as an acronym for Buchsbaum–Eisenbud–Fitting.
The fact that (2.15) may be infinite causes no problem, since, for degree reasons, \(U\) and \(R\) only contain a finite number of terms.
References
Andersson, M.: Residue currents and ideals of holomorphic functions. Bull. Sci. Math. 128, 481–512 (2004)
Andersson, M.: The membership problem for polynomial ideals in terms of residue currents. Ann. Inst. Fourier 56, 101–119 (2006)
Andersson, M.: Explicit versions of the Briançon–Skoda theorem with variations. Michigan Math. J. 54(2), 361–373 (2006)
Andersson, M.: A residue criterion for strong holomorphicity. Arkiv. Mat. 48, 1–15 (2010)
Andersson, M., Götmark, E.: Explicit representation of membership of polynomial ideals. Math. Ann. 349, 345–365 (2011)
Andersson, M., Samuelsson, H.: A Dolbeault–Grothendieck lemma on a complex space via Koppelman formulas. Invent. Math. 190, 261–297 (2012)
Andersson, M., Samuelsson, H., Sznajdman, J.: On the Briançon–Skoda theorem on a singular variety. Ann. Inst. Fourier 60, 417–432 (2010)
Andersson, M., Wulcan, E.: Residue currents with prescribed annihilator ideals. Ann. Sci. École Norm. Sup. 40, 985–1007 (2007)
Andersson, M., Wulcan, E.: Decomposition of residue currents. J. Reine Angew. Math. 638, 103–118 (2010)
Andersson, M., Wulcan, E.: On the effective membership problem on singular varieties. arXiv:1107.0388v2.
Björk, J.-B.: Residues and \({\cal D}\)-modules. The legacy of Niels Henrik Abel, 605651, Springer, Berlin (2004)
Björk, J.-E., Samuelsson, H.: Regularizations of residue currents. J. Reine Angew. Math. 649, 33–54 (2010)
Briançon, J., Skoda, H.: Sur la clôture intégrale d’un idéal de germes de fonctions holomorphes en un point de \({\mathbb{C}}^{n}\). C. R. Acad. Sci. Paris Sér. A 278, 949–951 (1974)
Brownawell, W.D.: Bounds for the degrees in the Nullstellensatz. Ann. Math. 126(3), 577–591 (1987)
Demailly, J.-P.: Complex Analytic and Differential Geometry. Monograph Grenoble (1997)
Ein, L., Lazarsfeld, R.: A geometric effective Nullstellensatz. Invent. Math. 135, 427–448 (1999)
Eisenbud, D.: Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, 160. Springer, New York (1995).
Eisenbud, D.: The geometry of syzygies. A second course in commutative algebra and algebraic geometry. Graduate Texts in Mathematics, 229. Springer, New York (2005).
Henkin, G., Passare, M.: Abelian differentials on singular varieties and variations on a theorem of Lie–Griffiths. Invent. Math. 135, 297–328 (1999)
Hermann, G.: Die Frage der endlich vielen Schritte in der Theorie der Polynomideale. Math. Ann. 95, 736–788 (1926)
Hickel, M.: Solution d’une conjecture de C. Berenstein-A. Yger et invariants de contact à l’infini. Ann. Inst. Fourier 51, 707–744 (2001)
Huneke, C.: Uniform bounds in Noetherian rings. Invent. Math. 107, 203–223 (1992)
Jelonek, Z.: On the effective Nullstellensatz. Invent. Math. 162, 1–17 (2005)
Kollár, J.: Sharp effective Nullstellensatz. J. Am. Math. Soc. 1, 963–975 (1988).
Lärkäng, R.: On the duality theorem on an analytic variety. Math. Ann. 355, 215–234 (2013)
Lazarsfeld, R.: Positivity in algebraic geometry I and II. Springer, Berlin (2004).
Macaulay, F.S.: The Algebraic Theory of Modular Systems. Cambridge Univ. Press, Cambridge (1916).
Mayr, E., Mayer, A.: The complexity of the word problem for commutative semigroups and polynomial ideals. Adv. Math. 46, 305–329 (1982).
Passare, M., Tsikh, A., Yger, A.: Residue currents of the Bochner–Martinelli type. Publicacions Mat. 44, 85–117 (2000)
Sombra, M.: A sparse effective Nullstellensatz. Adv. Appl. Math. 22, 271–295 (1999)
Sznajdman, J.: The Briançon-Skoda number of analytic irreducible planar curves. Ann. Inst. Fourier, to appear. Available at arXiv:1201.3288.
Wulcan, E.: Sparse effective membership problems via residue currents. Math. Ann. 350, 661–682 (2011)
Wulcan, E.: Some variants of Macaulay’s and Max Noether’s theorems. J. Commut. Algebra 2(4), 567–580 (2010)
Acknowledgments
We would like to thank the referee for careful reading and many valuable comments and suggestions that have substantially improved the exposition of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was partially supported by the Swedish Research Council. The second author was partially supported by the Swedish Research Council and by the NSF.
Rights and permissions
About this article
Cite this article
Andersson, M., Wulcan, E. Global effective versions of the Briançon–Skoda–Huneke theorem. Invent. math. 200, 607–651 (2015). https://doi.org/10.1007/s00222-014-0544-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-014-0544-x