Intersection theory on mixed curves

TL;DR

This paper extends the topological intersection theory for mixed curves in complex two-space to cases involving non-transversal intersections and singularities, using mixed polynomial definitions.

Contribution

It generalizes the intersection number definition for mixed curves to non-transversal and singular cases, expanding the applicability of the theory.

Findings

Defined intersection number for non-transversal intersections

Extended intersection theory to singular mixed curves

Provided a topological framework for mixed curve intersections

Abstract

We consider two mixed curve $C,C'\subset {\Bbb C}^2$ which are defined by mixed functions of two variables $\bf z=(z_1,z_2)$. We have shown in \cite{MC}, that they have canonical orientations. If $C$ and $C'$ are smooth and intersect transversely at $P$, the intersection number $I_{top}(C,C';P)$ is topologically defined. We will generalize this definition to the case when the intersection is not necessarily transversal or either $C$ or $C'$ may be singular at $P$ using the defining mixed polynomials.