Plane Curves of Minimal Degree with Prescribed Singularities

New Asymptotics for the Existence of Plane Curves with Prescribed Singularities

We study the classical problem of the existence of irreducible plane curves with given degree d and prescribed singularities of topological types S1 ; : : : ; Sr. Our main results concern substantial improvements of the leading coeecients in the asymptotically optimal suucient conditions for complex plane curves which were found in GLS1]. We then transfer these results to real plane curves.

An asymptotic existence theorem for plane curves with prescribed singularities

Let d,m1, . . . ,mr be (r + 1) positive integers. Denote by V (d;m1, . . . ,mr) the variety of irreducible (complex) plane curves of degree d having exactly r ordinary singularities of multiplicities m1, . . . ,mr. In most cases, it is still an open problem to know whether this variety is empty or not. In this paper, we will concentrate on the case where the r singularities can be taken in a ge...

Analytic order of singular and critical points

We deal with the following closely related problems: (i) For a germ of a reduced plane analytic curve, what is the minimal degree of an algebraic curve with a singular point analytically equivalent (isomorphic) to the given one? (ii) For a germ of a holomorphic function in two variables with an isolated critical point, what is the minimal degree of a polynomial, equivalent to the given function...

On the Existence of Plane Curves with Prescribed Singularities

On the Exceptional Locus of the Birational Projections of a Normal Surface Singularity into a Plane

Given a normal surface singularity (X,Q) and a birational morphism to a nonsingular surface π : X → S, we investigate the local geometry of the exceptional divisor L of π. We prove that the dimension of the tangent space to L at Q equals the number of exceptional components meeting at Q. Consequences relative to the existence of such birational projections contracting a prescribed number of irr...