Multiplier ideals are an interesting tool in singularity theory and birational geometry. From them one can get information on the type of singularity corresponding to an ideal, divisor or metric. They also accomplish several useful vanishing theorems.
This interesting paper considers multiplier ideals \(J(\xi C)_o\) corresponding to a plane curve singularity \((C,o)\) on a smooth complex algebraic surface \(S\) and a rational number \(\xi >0\) and the authors give a conceptually new description of these multiplier ideals (and their associated jumping numbers) in terms of a finite set of Newton polygons appearing in a toroidal embedded resolution process of the singularity.
Section 3.4 of the paper describes the construction of the above mentioned toroidal resolution following ideas of García Barroso, Popescu-Pampu and the first author generalizing processes previously introduced by several other authors.
Set \(\mathcal{C}\{x,y\}\) the local ring of germs of holomorphic functions of \(S\) at \(o\). The main results are Theorem 4.9 and 4.20. Roughly speaking, Theorem 4.9 states that \(J(\xi C)_o\) consists of the functions \(h \in \mathcal{C}\{x,y\}\) satisfying certain inclusions involving Newton polygons of the the singularities given by \(h\) and \(C\) with respect to some crosses which are pairs defined by germs of exceptional divisors and suitable curvettes intersecting them. Theorem 4.20 shows a way of regarding jumping numbers and multiplier ideals depending on monomial expressions.
From the results in the paper, it can be deduced that the computation of multiplier ideals and jumping numbers of \(C\) is reduced to an optimization problem in terms of log discrepancies of the rupture components of \(\pi*(C)\), \(\pi\) being a toroidal embedded resolution, and values attached to the corresponding exceptional divisors.