Containment problems for fat point schemes via asymptotic multiplier ideals

We study a multiplier-ideal approach to the containment problem for radical ideals of points in projective space. The expected asymptotic multiplier-ideal computation depends on controlling residual base loci of the relevant symbolic linear systems. We isolate this issue in the notion of clean asymptotic multiplier ideals. Under this hypothesis, a failure of $$I^{(2r-1)}\subseteq ̸I^r$$ forces $$2r\widehat{\alpha }(I)\ge \alpha (I^{(2r-1)})+3,$$ and in particular $\widehat{\alpha }(I)\ge 3$. We show that the dual-Hesse configuration is clean although its symbolic systems have a genuinely non-trivial residual asymptotic base part. We also verify cleanliness for star configurations of lines, giving a further infinite family of non-monomial examples, and formulate a cleanliness conjecture for singular loci of line arrangements.