×

Multiplier ideals of monomial ideals. (English) Zbl 0979.13026

In this paper the author treats the effective determination of multiplier ideals, a subproblem arising in the singularity analysis of divisors, ideal sheaves, or metrics. In particular, the multiplier ideals of monomial ideals are described as the integral grid points of shifts of the corresponding Newton polygon. The applicability of this characterization is demonstrated by several families of examples.
Reviewer: F.Winkler (Linz)

MSC:

13P05 Polynomials, factorization in commutative rings
14Q99 Computational aspects in algebraic geometry
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Urban Angehrn and Yum Tong Siu, Effective freeness and point separation for adjoint bundles, Invent. Math. 122 (1995), no. 2, 291 – 308. · Zbl 0847.32035 · doi:10.1007/BF01231446
[2] V. I. Arnol\(^{\prime}\)d, S. M. Guseĭn-Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. I, Monographs in Mathematics, vol. 82, Birkhäuser Boston, Inc., Boston, MA, 1985. The classification of critical points, caustics and wave fronts; Translated from the Russian by Ian Porteous and Mark Reynolds.
[3] Jean-Pierre Demailly, \?² vanishing theorems for positive line bundles and adjunction theory, Transcendental methods in algebraic geometry (Cetraro, 1994) Lecture Notes in Math., vol. 1646, Springer, Berlin, 1996, pp. 1 – 97. · Zbl 0883.14005 · doi:10.1007/BFb0094302
[4] Jean-Pierre Demailly and János Kollár. Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds. Mathematics E-Print Archive, October 1999. · Zbl 0994.32021
[5] Lawrence Ein, Multiplier ideals, vanishing theorems and applications, Algebraic geometry — Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 203 – 219. · Zbl 0978.14004
[6] Lawrence Ein and Robert Lazarsfeld, A geometric effective Nullstellensatz, Invent. Math. 137 (1999), no. 2, 427 – 448. · Zbl 0944.14003 · doi:10.1007/s002220050332
[7] Topics related to cohomologies of local rings, Kyoto University, Research Institute for Mathematical Sciences, Kyoto, 1985 (Japanese). Sūrikaisekikenkyūsho Kōkyūroku No. 543 (1985) (1985).
[8] William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. · Zbl 0813.14039
[9] János Kollár, Singularities of pairs, Algebraic geometry — Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 221 – 287. · Zbl 0905.14002
[10] Maclagan. Antichains of monomial ideals are finite. Mathematics E-Print Archive, October 1999. · Zbl 0984.13013
[11] V. V. Shokurov, Three-dimensional log perestroikas, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 1, 105 – 203 (Russian); English transl., Russian Acad. Sci. Izv. Math. 40 (1993), no. 1, 95 – 202. · Zbl 0785.14023 · doi:10.1070/IM1993v040n01ABEH001862
[12] Yum-Tong Siu, Invariance of plurigenera, Invent. Math. 134 (1998), no. 3, 661 – 673. · Zbl 0955.32017 · doi:10.1007/s002220050276
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.