An Algorithm for Computing the Singularity of the Generic Germ of a Pencil of Plane Curves

On adjacent complete ideals above a given complete ideal

“…Result 3.8 (theorem 2.5 of [1]). The cluster T equals the set of points q ∈ N O such that h q < w q .…”

“…For any q ∈ T , the virtual values are t q = w q and the virtual multiplicities are τ q = w q − h q . Result 3.9 (lemma 3.1 of [1]). If q belongs to T , then q is either satellite, or lies on ξ or on η.…”

“…Assign to each point q ∈ N O two integers w q = min{v q (ξ), v q (η)} and h q = q→s w s , h O = 0, the second one defined using induction on the order of neighbourhood that q belongs to. To find out the whole weighted cluster BP(f, g), we will use a pair of results given in [1] which we will now list, suitably adapted to the case that concerns us.…”

Effective computation of base points of ideals in two-dimensional local rings

“…Let K ≤p be the weighted subcluster consisting on all the points preceding a given point p ∈ K with the same weights as K. If we denote byξ p the virtual transform with respect to K ≤p , then, for all p ∈ K, we have 1 We use the notation E p and E p to emphasize that the exceptional component corresponds to the point p ∈ K.…”

“…It is a classical result that the equisingularity class, and equivalently the topological class [13], [5], of any element f = f α 1 1 · · · f αr r ∈ O X,O with f 1 , . .…”