Khovanskii-finite rational curves of arithmetic genus 2

TL;DR

This paper investigates conditions under which rational curves of genus two admit Khovanskii-finite valuations, providing explicit descriptions and methods to determine their existence, with implications for toric degenerations.

Contribution

It offers a semi-explicit description of genus two rational curves with Khovanskii-finite valuations and an effective method to verify this property over number fields.

Findings

Genus two rational curves with a single unibranched singularity are always Khovanskii-finite for small genus.

A criterion is established for the existence of Khovanskii-finite valuations on these curves.

The paper provides a practical method to decide if such curves admit toric degenerations.

Abstract

We study the existence of Khovanskii-finite valuations for rational curves of arithmetic genus two. We provide a semi-explicit description of the locus of degree $n+2$ rational curves in $\mathbb{P}^n$ of arithmetic genus two that admit a Khovanskii-finite valuation. Furthermore, we describe an effective method for determining if a rational curve of arithmetic genus two defined over a number field admits a Khovanskii-finite valuation. This provides a criterion for deciding if such curves admit a toric degeneration. Finally, we show that rational curves with a single unibranched singularity are always Khovanskii-finite if their arithmetic genus is sufficiently small.