A note on the multiplier ideals of monomial ideals

Gong, Cheng ; Tang, Zhongming

Czechoslovak Mathematical Journal, Tome 65 (2015) no. 4, pp. 905-913

Résumé

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 $0$. 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.

Détail

DOI : 10.1007/s10587-015-0216-z

Classification : 14F18
Keywords: multiplier ideal; monomial ideal; convex set