On the stability and regularity of the multiplier ideals of monomial ideals

Let a ⊆ ℂ[x1, . . . , xd] be a monomial ideal and J (a) its multiplier ideal which is also a monomial ideal. It is proved that if a is strongly stable or squarefree strongly stable then so is J (a). Denote the maximal degree of minimal generators of a by d(a). When a is strongly stable or squarefree strongly stable, it is shown that the Castelnuovo-Mumford regularity of J (a) is less than or equal to d(a). As a corollary, one gets a vanishing result on the ideal sheaf]J(a)˜\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde {\mathcal{J}\left( a \right)}$$\end{document} on ℙd–1 associated to J (a) that Hi(ℙd–1;J(a)˜\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde {\mathcal{J}\left( a \right)}$$\end{document}(s–i)) = 0, for all i > 0 and s ≥ d(a).