Real inflection points of real hyperelliptic curves

Abstract

Given a real hyperelliptic algebraic curve $X$ with non-empty real part and a real effective divisor $\mc{D}$ arising via pullback from ${\mathbb{P}}^1$ under the hyperelliptic structure map, we study the real inflection points of the associated complete real linear series $|\mc{D}|$ on $X$. To do so we use Viro's patchworking of real plane curves, recast in the context of some Berkovich spaces studied by M. Jonsson. Our method gives a simpler and more explicit alternative to limit linear series on metrized complexes of curves, as developed by O. Amini and M. Baker, for curves embedded in toric surfaces.

arXiv commentv.2: improved exposition following the referee's suggestions, and included a new result (Thm 6.1) giving a more refined construction of real inflection on complete series on hyperelliptic curves that includes non-maximal cases. To appear in Transactions of the AMS