Classification of normal phylogenetic varieties for tripods

TL;DR

This paper classifies normal phylogenetic varieties derived from tripods and trivalent trees, establishing conditions for projective normality based on the underlying abelian group structure.

Contribution

It provides a complete classification of when group-based phylogenetic varieties from trivalent trees are projectively normal, identifying specific abelian groups that satisfy this property.

Findings

Group-based phylogenetic varieties are projectively normal iff G is in {Z_2, Z_3, Z_2×Z_2, Z_4, Z_5, Z_7}.

Complete classification for normality of phylogenetic varieties from tripods and trivalent trees.

Advances understanding of algebraic properties of phylogenetic models in evolutionary biology.

Abstract

We provide a complete classification of normal phylogenetic varieties coming from tripods, and more generally, from trivalent trees. Let $G$ be an abelian group. We prove that the group-based phylogenetic variety $X_{G,\mathcal{T}}$, for any trivalent tree $\mathcal{T}$, is projectively normal if and only if $G\in \{\mathbb{Z}_2, \mathbb{Z}_3, \mathbb{Z}_2\times\mathbb{Z}_2, \mathbb{Z}_4, \mathbb{Z}_5, \mathbb{Z}_7\}$.