Encomplexed Brown Invariant of Real Algebraic Surfaces in RP^3

TL;DR

This paper introduces a new invariant for parametrized real algebraic surfaces in RP^3, extending the classical Brown invariant from smooth topology to algebraic settings using self intersection properties.

Contribution

It generalizes the Brown invariant to real algebraic surfaces in RP^3 by defining a self linking number for their self intersection curves.

Findings

Constructed a new invariant for real algebraic surfaces in RP^3.

Extended the concept of self linking number to algebraic surface self intersections.

Provided a framework connecting algebraic geometry with topological invariants.

Abstract

We construct an invariant of parametrized generic real algebraic surfaces in RP^3 which generalizes the Brown invariant of immersed surfaces from smooth topology. The invariant is constructed using the self intersection, which is a real algebraic curve with points of three local characters: the intersection of two real sheets, the intersection of two complex conjugate sheets or a Whitney umbrella. The Brown invariant was expressed through a self linking number of the self intersection by Kirby and Melvin. We extend the definition of this self linking number to the case of parametrized generic real algebraic surfaces.