An effective avoidance principle for a class of ideals (Q1745313)

Let \(S\) be a polynomial ring over a field of characteristic zero and let \(I\subset S\) be a monomial ideal. The ideal \(I\) is said to be of intersection type if it can be written as the intersection of powers of monomial prime ideals. This kind of ideals was introduced by \textit{J. Herzog} and \textit{M. Vladoiu} [Electron. J. Comb. 21, No. 1, Research Paper P1.69, 18 p. (2014; Zbl 1307.13014)]. In the paper under review, the author investigates the ideals \(I\) which are of intersection type and moreover, have no embedded primary component. The main goal is to provide an effective sufficient condition for a given monomial prime ideal to avoid the sets of prime divisors of the powers of \(I\), and in particular to avoid the celebrated set of asymptotic prime divisors of \(I\), which will follow from a new and quite surprising double-colon stability property. Further, the author briefly describes some other applications, e.g., on the topology of a suitable blowing-up.