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 \(\widetilde {J\left( a \right)}\) on ℙd−1 associated to J (a) that \({H^i}\left( {{P^{d - 1}};\widetilde {J\left( a \right)}\left( {s - i} \right)} \right) = 0\), for all i > 0 and s ≥ d(a).