Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 42, No. 2, pp. 509-516 (2001)

Inflection Points on Real Plane Curves Having Many Pseudo-Lines

Johannes Huisman

Institut Mathématique de Rennes, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France, e-mail:

Abstract: \font\msbm=msbm10 \def\P{\hbox{\msbm P}} \def\R{\hbox{\msbm R}} A pseudo-line of a real plane curve $C$ is a global real branch of $C(\R)$ that is not homologically trivial in $\P^2(\R)$. A geometrically integral real plane curve $C$ of degree $d$ has at most $d-2$ pseudo-lines, provided that $C$ is not a real projective line. Let $C$ be a real plane curve of degree $d$ having exactly $d-2$ pseudo-lines. Suppose that the genus of the normalization of $C$ is equal to $d-2$. We show that each pseudo-line of $C$ contains exactly $3$ inflection points. This generalizes the fact that a nonsingular real cubic has exactly $3$ real inflection points.

Keywords: real plane curve, pseudo-line, inflection point

Classification (MSC2000): 14H45, 14P99

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