One-forms on singular curves and the topology of real curve singularities

Abstract

sense the zeros and poles of the form respectively. These modules are finite dimensional vector spaces, and we prove in $1 that the difference in dimension is preserved under deformation of both the form and the curve. In the case that r = dg for some holomorphic function g, this enables us to find the number of critical points of a small generic deformation ofg. Further, we give a simple proof of the fact that the jump in Milnor number (as defined in [3] and [6]) in a flat family of curve singularities is equal to the vanishing Euler characteristic. In the case that V and a are real, the l-form defines an orientation on each connected component of V - {p} ( = half-branch), where p is the base point of %, Some of these halfbranches will be oriented outwards and some inwards. Moreover, the two ramification modules come with real valued non-degenerate quadratic forms. We show in $2 that the sum of signatures of these two forms is equal to the difference between the numbers of branches oriented outwards and those oriented inwards. This is related to the classical method of Hermite for calculating the number of real roots of a polynomial as the signature of a quadratic form (see[12]). We remark that it seems surprising that the two important features of these ramification modules are the difference of the dimensions, but the sum of the signatures.