On the variety of plane curves of degree d with a singular point of multiplicity m

Abstract

AbstractWe study a compactification of the variety Ud,m, 1<m<d, of plane curves of degree d with an ordinary singular point of multiplicity m as a unique singularity, given by means of a projective bundle Xd,m. We show that the boundary Xd,m-Ud,m consists of two irreducible components of codimension 1: Ad,m, the closure of the variety of curves with two singular points, one of multiplicity m and another of multiplicity 2, and Bd,m, the closure of the variety of curves with a non-ordinary singular point of multiplicity m. We determine the relations that express the classes of Ad,m and Bd,m in terms of a basis of the group Pic(Xd,m). From this we describe the Picard group of the variety Ud,m, obtaining that it is a finite group of order 3(m-1)(d-m)[2d2-(m+4)d-(m2-2m-2)]