DEFORMATION OF COMPLETE INTERSECTIONS IN THE PLANE

Abstract

We study deformations of zero-dimensional complete intersections in the plane, and prove the following results. (1) Two complex non-singular curves intersecting at r points with multiplicities d₁, …, d_r can be deformed into curves intersecting (at some points) with multiplicities d′₁, …, d′_s which are arbitrary prescribed partitions of the numbers d₁, …, d_r. (2) Two real curves intersecting with multiplicity at most 2 at each of their real common points can be deformed so that all real multiple intersection points split into real simple intersection points.