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A short note on the multiplier ideals of monomial space curves. (English) Zbl 1342.14044

Multiplier ideals are known as an important tool in the study of birational geometry, especially when they appear in vanishing theorems. It is interesting to know how to calculate multiplier ideals in some concrete examples. The possibly most well-known result is Howard’s theorem which gives a nice formula for multiplier ideals of monomial ideals in terms of Newton polyhedrons. The paper under review concerns the calculation of multiplier ideals for monomial space curves. The main theorem improves a former result obtained by the author [Proc. Am. Math. Soc., Ser. B 1, 33–41 (2014; Zbl 1339.14016)]. The idea is to use auxiliary valuations on the lattice. The former result claims that the number of valuations needed is finite. The author improves it so that we only need two valuations.

MSC:

14F18 Multiplier ideals
14H50 Plane and space curves
14M25 Toric varieties, Newton polyhedra, Okounkov bodies

Citations:

Zbl 1339.14016
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References:

[1] Alberich-Carramiñana, M.; Àlvarez Montaner, J.; Dachs-Cadefau, F., Multiplier ideals in two-dimensional local rings with rational singularities (2014), 32 pp. · Zbl 1357.14025
[2] Blanco, R.; Encinas, S., A procedure for computing the log canonical threshold of a binomial ideal (2014), 28 pp. · Zbl 1422.14027
[3] Bravo, A., A remark on strong factorizing resolutions, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., 107, 1, 53-60 (2013), MR3031261 · Zbl 1258.14017
[4] Blickle, M., Multiplier ideals and modules on toric varieties, Math. Z., 248, 1, 113-121 (2004), MR2092724 (2006a:14082) · Zbl 1061.14055
[5] Eisenstein, E., Generalizations of the restriction theorem for multiplier ideals (2010), 17 pp.
[6] Galindo, C.; Monserrat, F., The Poincaré series of multiplier ideals of a simple complete ideal in a local ring of a smooth surface, Adv. Math., 225, 2, 1046-1068 (2010), MR2671187 (2012a:14039) · Zbl 1206.14011
[7] González Pérez, P. D.; Teissier, B., Embedded resolutions of non necessarily normal affine toric varieties, C. R. Math. Acad. Sci. Paris, 334, 5, 379-382 (2002), (English, with English and French summaries). MR1892938 (2003b:14019) · Zbl 1052.14062
[8] Howald, J. A., Multiplier ideals of monomial ideals, Trans. Am. Math. Soc., 353, 7, 2665-2671 (2001), (electronic). MR1828466 (2002b:14061) · Zbl 0979.13026
[9] Howald, J. A., Multiplier ideals of sufficiently general polynomials (2003), 9 pp.
[10] Hyry, E.; Järvilehto, T., Jumping numbers and ordered tree structures on the dual graph, Manuscr. Math., 136, 3-4, 411-437 (2011), MR2844818 · Zbl 1235.13018
[11] Naie, D., Jumping numbers of a unibranch curve on a smooth surface, Manuscr. Math., 128, 1, 33-49 (2009), 2470185 (2009j:14034) · Zbl 1165.14025
[12] Naie, D., Mixed multiplier ideals and the irregularity of abelian coverings of smooth projective surfaces, Expo. Math., 31, 1, 40-72 (2013), MR3035120 · Zbl 1272.14015
[13] Shibuta, T.; Takagi, S., Log canonical thresholds of binomial ideals, Manuscr. Math., 130, 1, 45-61 (2009), MR2533766 · Zbl 1183.13007
[14] Smith, K. E.; Thompson, H. M., Irrelevant exceptional divisors for curves on a smooth surface, (Algebra, Geometry and their Interactions. Algebra, Geometry and their Interactions, Contemp. Math., vol. 448 (2007), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 245-254, MR2389246 (2009c:14004) · Zbl 1141.14004
[15] Teitler, Z., Software for multiplier ideals (2013), 7 pp.
[16] Thompson, H. M., Comments on toric varieties (2003), 6 pp.
[17] Thompson, H. M., Multiplier ideals of monomial space curves, Proc. Am. Math. Soc. Ser. B, 1, 33-41 (2014), MR3168880 · Zbl 1339.14016
[18] Tucker, K., Jumping numbers on algebraic surfaces with rational singularities, Trans. Am. Math. Soc., 362, 6, 3223-3241 (2010), MR2592954 (2011c:14106) · Zbl 1194.14026
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