| Title:
|
A note on the multiplier ideals of monomial ideals (English) |
| Author:
|
Gong, Cheng |
| Author:
|
Tang, Zhongming |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
65 |
| Issue:
|
4 |
| Year:
|
2015 |
| Pages:
|
905-913 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
multiplier ideal |
| Keyword:
|
monomial ideal |
| Keyword:
|
convex set |
| MSC:
|
14F18 |
| idZBL:
|
Zbl 06537699 |
| idMR:
|
MR3441324 |
| DOI:
|
10.1007/s10587-015-0216-z |
| . |
| Date available:
|
2016-01-13T09:03:41Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144781 |
| . |
| Reference:
|
[1] Blickle, M.: Multiplier ideals and modules on toric varieties.Math. Z. 248 (2004), 113-121. Zbl 1061.14055, MR 2092724, 10.1007/s00209-004-0655-y |
| Reference:
|
[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. Zbl 1084.14015, MR 2132649 |
| Reference:
|
[3] Demailly, J.-P., Ein, L., Lazarsfeld, R.: A subadditivity property of multiplier ideals.Mich. Math. J. 48 (2000), 137-156. Zbl 1077.14516, MR 1786484, 10.1307/mmj/1030132712 |
| Reference:
|
[4] Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry.Graduate Texts in Mathematics 150 Springer, Berlin (1995). Zbl 0819.13001, MR 1322960 |
| Reference:
|
[5] Esnault, H., Viehweg, E.: Lectures on Vanishing Theorems.DMV Seminar 20 Birkhäuser, Basel (1992). Zbl 0779.14003, MR 1193913 |
| Reference:
|
[6] Fulton, W.: Introduction to Toric Varieties.Annals of Mathematics Studies 131 Princeton University Press, Princeton (1993). Zbl 0813.14039, MR 1234037 |
| Reference:
|
[7] Hara, N., Yoshida, K.-I.: A generalization of tight closure and multiplier ideals.Trans. Am. Math. Soc. 355 (2003), 3143-3174. Zbl 1028.13003, MR 1974679, 10.1090/S0002-9947-03-03285-9 |
| Reference:
|
[8] Hartshorne, R.: Algebraic Geometry.Graduate Texts in Mathematics 52 Springer, New York (1977). Zbl 0367.14001, MR 0463157 |
| Reference:
|
[9] Howald, J. A.: Multiplier ideals of monomial ideals.Trans. Am. Math. Soc. 353 (2001), 2665-2671. Zbl 0979.13026, MR 1828466, 10.1090/S0002-9947-01-02720-9 |
| Reference:
|
[10] Hübl, R., Swanson, I.: Adjoints of ideals.Mich. Math. J. 57 (2008), 447-462. Zbl 1180.13005, MR 2492462, 10.1307/mmj/1220879418 |
| Reference:
|
[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 |
| Reference:
|
[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. Zbl 0796.13020, MR 1316244 |
| Reference:
|
[13] Nadel, A. M.: Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature.Ann. Math. (2) 132 (1990), 549-596. Zbl 0731.53063, MR 1078269 |
| Reference:
|
[14] Siu, Y.-T.: Multiplier ideal sheaves in complex and algebraic geometry.Sci. China, Ser. A 48 (2005), 1-31. Zbl 1131.32010, MR 2156488, 10.1007/BF02884693 |
| Reference:
|
[15] Swanson, I., Huneke, C.: Integral Closure of Ideals, Rings, and Modules.London Mathematical Society Lecture Note Series 336 Cambridge University Press, Cambridge (2006). Zbl 1117.13001, MR 2266432 |
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