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A note on the multiplier ideals of monomial ideals. (English) Zbl 1363.14001

Summary: Let \(\mathfrak{a}\subseteq \mathbb{C}[x_1,\ldots ,x_n]\) be a monomial ideal and \(\mathcal{J}(\mathfrak{a}^c)\) the multiplier ideal of \(\mathfrak{a}\) with coefficient \(c\). Then \(\mathcal{J}(\mathfrak{a}^c)\) is also a monomial ideal of \(\mathbb{C}[x_1,\ldots ,x_n]\), and the equality \(\mathcal{J}(\mathfrak{a}^c)=\mathfrak{a}\) implies that \(0<c<n+1\). We mainly discuss the problem when \(\mathcal{J}(\mathfrak{a})=\mathfrak{a}\) or \(\mathcal{J}(\mathfrak{a}^{n+1-\varepsilon})=\mathfrak{a}\) for all \(0<\varepsilon <1\). It is proved that if \(\mathcal{J}(\mathfrak{a})=\mathfrak{a}\) then \(\mathfrak{a}\) is principal, and if \(\mathcal{J}(\mathfrak{a}^{n+1-\varepsilon})=\mathfrak{a}\) holds for all \(0<\varepsilon <1\) then \(\mathfrak{a}=(x_1,\ldots ,x_n)\). One global result is also obtained. Let \(\tilde {\mathfrak{a}}\) be the ideal sheaf on \(\mathbb{P}^{n-1}\) associated with \(\mathfrak{a}\). Then it is proved that the equality \(\mathcal{J}(\tilde {\mathfrak{a}})=\tilde {\mathfrak{a}}\) implies that \(\tilde {\mathfrak{a}}\) is principal.

MSC:

14F18 Multiplier ideals
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
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References:

[1] M. Blickle: Multiplier ideals and modules on toric varieties. Math. Z. 248 (2004), 113–121. · Zbl 1061.14055 · doi:10.1007/s00209-004-0655-y
[2] M. Blickle, R. Lazarsfeld: An informal introduction to multiplier ideals. Trends in Commutative Algebra. (L. L. Avramov et al., eds.). Mathematical Sciences Research Institute Publications 51, Cambridge University Press, Cambridge, 2004, pp. 87–114. · Zbl 1084.14015
[3] J.-P. Demailly, L. Ein, R. Lazarsfeld: A subadditivity property of multiplier ideals. Mich. Math. J. 48 (2000), 137–156. · Zbl 1077.14516 · doi:10.1307/mmj/1030132712
[4] D. Eisenbud: Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics 150, Springer, Berlin, 1995. · Zbl 0819.13001
[5] H. Esnault, E. Viehweg: Lectures on Vanishing Theorems. DMV Seminar 20, Birkhäuser, Basel, 1992. · Zbl 0779.14003
[6] W. Fulton: Introduction to Toric Varieties. Annals of Mathematics Studies 131, Princeton University Press, Princeton, 1993. · Zbl 0813.14039
[7] N. Hara, K.-I. Yoshida: A generalization of tight closure and multiplier ideals. Trans. Am. Math. Soc. 355 (2003), 3143–3174. · Zbl 1028.13003 · doi:10.1090/S0002-9947-03-03285-9
[8] R. Hartshorne: Algebraic Geometry. Graduate Texts in Mathematics 52, Springer, New York, 1977. · Zbl 0367.14001
[9] J. A. Howald: Multiplier ideals of monomial ideals. Trans. Am. Math. Soc. 353 (2001), 2665–2671. · Zbl 0979.13026 · doi:10.1090/S0002-9947-01-02720-9
[10] R. Hübl, I. Swanson: Adjoints of ideals. Mich. Math. J. 57 (2008), 447–462. · Zbl 1180.13005 · doi:10.1307/mmj/1220879418
[11] R. Lazarsfeld: Positivity in Algebraic Geometry II. Positivity for Vector Bundles, and Multiplier Ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge 49, Springer, Berlin, 2004. · Zbl 1093.14500
[12] J. Lipman: Adjoints and polars of simple complete ideals in two-dimensional regular local rings. Bull. Soc. Math. Belg., Sér. A 45 (1993), 223–244. · Zbl 0796.13020
[13] A. M. Nadel: Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature. Ann. Math. (2) 132 (1990), 549–596. · Zbl 0731.53063 · doi:10.2307/1971429
[14] Y.-T. Siu: Multiplier ideal sheaves in complex and algebraic geometry. Sci. China, Ser. A 48 (2005), 1–31. · Zbl 1131.32010 · doi:10.1007/BF02884693
[15] I. Swanson, C. Huneke: Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series 336, Cambridge University Press, Cambridge, 2006. · Zbl 1117.13001
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