Toric geometry of the 3-Kimura model for any tree

TL;DR

This paper explores the geometric structure of the 3-Kimura model across any tree, providing a detailed description of the associated variety on a biologically relevant open set, and addressing a conjecture on ideal generation degree.

Contribution

It offers a precise geometric description of the 3-Kimura model's variety for any tree, advancing understanding of its algebraic properties and conjectures.

Findings

Describes the variety on a Zariski open set containing all biologically meaningful points

Provides insights into the ideal's generation degree for the 3-Kimura model

Addresses a conjecture by Sturmfels and Sullivant

Abstract

In this paper we present geometric features of group based models. We focus on the 3-Kimura model. We present a precise geometric description of the variety associated to any tree on a Zariski open set. In particular this set contains all biologically meaningful points. Our motivation is a conjecture of Sturmfels and Sullivant on the degree in which the ideal associated to 3-Kimura model is generated.