A formula for the degree of singularity of plane algebroid curves

Abstract

Let Of be the local ring of an algebroid reduced curve {f = 0} over an algebraically closed field K, Of its integral closure in the total quotient ring of Of and Cf the conductor of Of in Of . The codimension c(f) = dimK Of /Cf is called the degree of singularity of the curve {f = 0}. Suppose that the Newton polygon Nf of the curve {f = 0} intersects the axes at the points (m, 0),(0, n) and put c(Nf ) = 2, (area of the polygon bounded by Nf and the axes) + (number of integer points on Nf ) − m − n − 1. We prove that there exists a factorization f = f1 · · · fs of f in K[[x, y]] such that c(f) = c(Nf) +Ps i=1 c(˜fi), where {˜fi = 0} is obtained as a composition of quadratic transforms of the curve {fi = 0}. The proof is effective: the Newton polygon Nf and the initial parts of f corresponding to the compact edges of Nf determine the Newton polygons of fi and the number of quadratic transforms necessary to compute ˜fi. As application of our result we give a formula for the Milnor number of f.