A note on the multiplier ideals of monomial ideals

Cheng Gong; Zhongming Tang

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 4, page 905-913
  • ISSN: 0011-4642

Abstract

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Let 𝔞 [ x 1 , ... , x n ] be a monomial ideal and 𝒥 ( 𝔞 c ) the multiplier ideal of 𝔞 with coefficient c . Then 𝒥 ( 𝔞 c ) is also a monomial ideal of [ x 1 , ... , x n ] , and the equality 𝒥 ( 𝔞 c ) = 𝔞 implies that 0 < c < n + 1 . We mainly discuss the problem when 𝒥 ( 𝔞 ) = 𝔞 or 𝒥 ( 𝔞 n + 1 - ε ) = 𝔞 for all 0 < ε < 1 . It is proved that if 𝒥 ( 𝔞 ) = 𝔞 then 𝔞 is principal, and if 𝒥 ( 𝔞 n + 1 - ε ) = 𝔞 holds for all 0 < ε < 1 then 𝔞 = ( x 1 , ... , x n ) . One global result is also obtained. Let 𝔞 ˜ be the ideal sheaf on n - 1 associated with 𝔞 . Then it is proved that the equality 𝒥 ( 𝔞 ˜ ) = 𝔞 ˜ implies that 𝔞 ˜ is principal.

How to cite

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Gong, Cheng, and Tang, Zhongming. "A note on the multiplier ideals of monomial ideals." Czechoslovak Mathematical Journal 65.4 (2015): 905-913. <http://eudml.org/doc/276091>.

@article{Gong2015,
abstract = {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.},
author = {Gong, Cheng, Tang, Zhongming},
journal = {Czechoslovak Mathematical Journal},
keywords = {multiplier ideal; monomial ideal; convex set},
language = {eng},
number = {4},
pages = {905-913},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on the multiplier ideals of monomial ideals},
url = {http://eudml.org/doc/276091},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Gong, Cheng
AU - Tang, Zhongming
TI - A note on the multiplier ideals of monomial ideals
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 4
SP - 905
EP - 913
AB - 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.
LA - eng
KW - multiplier ideal; monomial ideal; convex set
UR - http://eudml.org/doc/276091
ER -

References

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  1. Blickle, M., 10.1007/s00209-004-0655-y, Math. Z. 248 (2004), 113-121. (2004) Zbl1061.14055MR2092724DOI10.1007/s00209-004-0655-y
  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. (2004) Zbl1084.14015MR2132649
  3. Demailly, J.-P., Ein, L., Lazarsfeld, R., 10.1307/mmj/1030132712, Mich. Math. J. 48 (2000), 137-156. (2000) Zbl1077.14516MR1786484DOI10.1307/mmj/1030132712
  4. Eisenbud, D., Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics 150 Springer, Berlin (1995). (1995) Zbl0819.13001MR1322960
  5. Esnault, H., Viehweg, E., Lectures on Vanishing Theorems, DMV Seminar 20 Birkhäuser, Basel (1992). (1992) Zbl0779.14003MR1193913
  6. Fulton, W., Introduction to Toric Varieties, Annals of Mathematics Studies 131 Princeton University Press, Princeton (1993). (1993) Zbl0813.14039MR1234037
  7. Hara, N., Yoshida, K.-I., 10.1090/S0002-9947-03-03285-9, Trans. Am. Math. Soc. 355 (2003), 3143-3174. (2003) Zbl1028.13003MR1974679DOI10.1090/S0002-9947-03-03285-9
  8. Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics 52 Springer, New York (1977). (1977) Zbl0367.14001MR0463157
  9. Howald, J. A., 10.1090/S0002-9947-01-02720-9, Trans. Am. Math. Soc. 353 (2001), 2665-2671. (2001) Zbl0979.13026MR1828466DOI10.1090/S0002-9947-01-02720-9
  10. Hübl, R., Swanson, I., 10.1307/mmj/1220879418, Mich. Math. J. 57 (2008), 447-462. (2008) Zbl1180.13005MR2492462DOI10.1307/mmj/1220879418
  11. Lazarsfeld, R., Positivity in Algebraic Geometry II. Positivity for Vector Bundles, and Multiplier Ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge Springer, Berlin (2004). (2004) MR2095471
  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. (1993) Zbl0796.13020MR1316244
  13. Nadel, A. M., Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature, Ann. Math. (2) 132 (1990), 549-596. (1990) Zbl0731.53063MR1078269
  14. Siu, Y.-T., 10.1007/BF02884693, Sci. China, Ser. A 48 (2005), 1-31. (2005) Zbl1131.32010MR2156488DOI10.1007/BF02884693
  15. Swanson, I., Huneke, C., Integral Closure of Ideals, Rings, and Modules, London Mathematical Society Lecture Note Series 336 Cambridge University Press, Cambridge (2006). (2006) Zbl1117.13001MR2266432

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