A sandwiched singularity \(X\) of a surface is a normal surface singularity which dominates birationally a non singular surface \(S\). These singularities are obtained from the basis \(S\) by blowing-up a complete \(\mathfrak M\)-primary ideal \(I\) of the local ring \({\mathcal O}_{S,O}\), \(S\) is a complex regular analytic surface. The usual problem in this theory is to get all possible information from \(I\subset {\mathcal O}_{S,O}\) on the singularities of \(X\). Here the informations given by the author are the number of singularities on \(X\), their fundamental cycles and multiplicities.
The main theorem 3.5 gives, in the more general case where \({\mathcal O}_{S,O}\) is a local \(\mathbb C\)-algebra having a rational singularity, a bijection between the set of complete ideals of codimension 1 (as \(\mathbb C\)-vector space) contained in \(I\) and the set of points in the exceptional locus of the surface \(X=\text{Bl}_I(R)\).
In the section 4, the author applies his theorem to the case where \(S\) is regular and \(X\) a sandwiched singularity. To every exceptional point \(Q\in X\), there is the associated ideal \(I_Q\) of 3.5 and to \(J\) a complete \(\mathfrak M\)-primary ideal, the author associates the weighted cluster of base points of \(J\) (base points are closed points in \(X'\) where general elements of \(J\) go through, \(X'\to X\) is any sequence of blow-ups of closed points).
All this leads to an explicit formula in theorem 4.7, which gives the multiplicity of an exceptional point \(Q\in X\) in terms of the clusters given by the base points of \(I\) and of \(I_Q\).