A note on the multiplier ideals of monomial ideals

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 -