On Huisman's conjectures about unramified real curves

TL;DR

This paper investigates Huisman's conjectures on the topology and classification of unramified real algebraic curves in projective space, providing counterexamples to one conjecture and insights into the other.

Contribution

The paper disproves Huisman's conjecture that unramified real curves in odd-dimensional projective space are M-curves with pseudo-line real branches, and discusses the validity of the second conjecture.

Findings

Counterexamples to Huisman's first conjecture are constructed.

The second conjecture holds for generic curves of odd degree based on inflection point formulas.

Abstract

Let $X \subset \mathbb{P}^{n}$ be an unramified real curve with $X(\mathbb{R}) \neq \emptyset$. If $n \geq 3$ is odd, Huisman conjectures that $X$ is an $M$-curve and that every branch of $X(\mathbb{R})$ is a pseudo-line. If $n \geq 4$ is even, he conjectures that $X$ is a rational normal curve or a twisted form of a such. We disprove the first conjecture by giving a family of counterexamples. We remark that the second conjecture follows for generic curves of odd degree from the formula enumerating the number of complex inflection points.