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
Full Text:

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
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.