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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.
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