On Characteristic Cones, Clusters and Chains of Infinitely Near Points

Let π : Z → X be a projective birational morphism of smooth surfaces. Assume π is an isomorphism outside π−1(Q) for some closed point Q ∊ X. For each factorization Z → Y → X, where Y is a normal surface, one has that Y is the blowing up of a complete (i.e. integrally closed) ideal I on X with IOZ invertible and support at Q. Since π is the composition of the successive blowing ups of a finite set C of infinitely near points (we will call C a constellation) to Q, the theorem of Zariski on unique factorization of complete ideals allow to describe all these sandwiched surfaces Y as well as the contractions Z → Y. Such a contraction becomes the minimal resolution of the singularities of Y (a class of rational singularities called sandwiched).