# Article

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Keywords:
multiplier ideal; monomial ideal; convex set
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 $\tilde {\frak {a}}$ be the ideal sheaf on ${\mathbb P}^{n-1}$ associated with $\frak {a}$. Then it is proved that the equality ${\mathcal J}(\tilde {\mathfrak {a}})=\tilde {\mathfrak {a}}$ implies that $\tilde {\mathfrak {a}}$ is principal.
References:
[1] Blickle, M.: Multiplier ideals and modules on toric varieties. Math. Z. 248 (2004), 113-121. DOI 10.1007/s00209-004-0655-y | MR 2092724 | Zbl 1061.14055
[2] Blickle, M., Lazarsfeld, R.: An informal introduction to multiplier ideals. Trends in Commutative Algebra. Mathematical Sciences Research Institute Publications 51 Cambridge University Press, Cambridge (2004), 87-114 L. L. Avramov et al. MR 2132649 | Zbl 1084.14015
[3] Demailly, J.-P., Ein, L., Lazarsfeld, R.: A subadditivity property of multiplier ideals. Mich. Math. J. 48 (2000), 137-156. DOI 10.1307/mmj/1030132712 | MR 1786484 | Zbl 1077.14516
[4] Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics 150 Springer, Berlin (1995). MR 1322960 | Zbl 0819.13001
[5] Esnault, H., Viehweg, E.: Lectures on Vanishing Theorems. DMV Seminar 20 Birkhäuser, Basel (1992). MR 1193913 | Zbl 0779.14003
[6] Fulton, W.: Introduction to Toric Varieties. Annals of Mathematics Studies 131 Princeton University Press, Princeton (1993). MR 1234037 | Zbl 0813.14039
[7] Hara, N., Yoshida, K.-I.: A generalization of tight closure and multiplier ideals. Trans. Am. Math. Soc. 355 (2003), 3143-3174. DOI 10.1090/S0002-9947-03-03285-9 | MR 1974679 | Zbl 1028.13003
[8] Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics 52 Springer, New York (1977). MR 0463157 | Zbl 0367.14001
[9] Howald, J. A.: Multiplier ideals of monomial ideals. Trans. Am. Math. Soc. 353 (2001), 2665-2671. DOI 10.1090/S0002-9947-01-02720-9 | MR 1828466 | Zbl 0979.13026
[10] Hübl, R., Swanson, I.: Adjoints of ideals. Mich. Math. J. 57 (2008), 447-462. DOI 10.1307/mmj/1220879418 | MR 2492462 | Zbl 1180.13005
[11] Lazarsfeld, R.: Positivity in Algebraic Geometry I\kern-1ptI. Positivity for Vector Bundles, and Multiplier Ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge {\it 49} Springer, Berlin (2004). MR 2095471
[12] Lipman, J.: Adjoints and polars of simple complete ideals in two-dimensional regular local rings. Bull. Soc. Math. Belg., Sér. A 45 (1993), 223-244. MR 1316244 | Zbl 0796.13020
[13] Nadel, A. M.: Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature. Ann. Math. (2) 132 (1990), 549-596. MR 1078269 | Zbl 0731.53063
[14] Siu, Y.-T.: Multiplier ideal sheaves in complex and algebraic geometry. Sci. China, Ser. A 48 (2005), 1-31. DOI 10.1007/BF02884693 | MR 2156488 | Zbl 1131.32010
[15] Swanson, I., Huneke, C.: Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series 336 Cambridge University Press, Cambridge (2006). MR 2266432 | Zbl 1117.13001

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