Algebraically unrealizable complex orientations of plane real pseudoholomorphic curves

TL;DR

This paper establishes inequalities for complex orientations of certain real algebraic curves and constructs pseudoholomorphic curves that violate these inequalities, showing their unoriented isotopy types are not realizable algebraically.

Contribution

It introduces new inequalities for Type I real algebraic curves and demonstrates their limitations through explicit pseudoholomorphic curve constructions.

Findings

Proves two inequalities for complex orientations of Type I curves.

Constructs pseudoholomorphic curves violating these inequalities.

Shows unoriented isotopy types are algebraically unrealizable.

Abstract

We prove two inequalities for the complex orientations of a separating (Type I) non-singular real algebraic curve in $RP^2$ of any odd degree. We also construct a separating non-singular pseudoholomorphic curve in $RP^2$ of any degree congruent to 9 mod 12 which does not satisfies one of these inequalities. Therefore the oriented isotopy type of the real locus of each of these curves is algebraically unrealizable.