A note on the multiplier ideals of monomial ideals. (English) Zbl 1363.14001

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 \(\tilde {\mathfrak{a}}\) 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.


14F18 Multiplier ideals
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
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