2007/02/27 by Marta Casanellas, Casanellas, Marta, Jesús Fernández-Sánchez +2 · 1 citation
Agricultural and Biological Sciences · Biochemistry, Genetics and Molecular Biology · Mathematics · Pharmacology, Toxicology and Pharmaceutics · #05C85 #14J99 #92D15 #Algebraic Geometry (math.AG) #Alkaloids: synthesis and pharmacology #Commutative Algebra (math.AC) #FOS: Biological sciences #FOS: Mathematics #Plant Diversity and Evolution #Plant and Fungal Species Descriptions #Populations and Evolution (q-bio.PE) #math.AC #math.AG #msc:05C85 #msc:14J99 #msc:92D15 #q-bio.PE
paper · pdf · doi:10.48550/arxiv.math/0702834
26 pages with 4 figures
arxiv created 2007/02/27 · openalex publication_date 2007/02/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Kimura 3-parameter model on a tree of n leaves is one of the most used in phylogenetics. The affine algebraic variety W associated to it is a toric variety. We study its geometry and we prove that it is isomorphic to a geometric quotient of the affine space by a finite group acting on it. As a consequence, we are able to study the singularities of W and prove that the biologically meaningful points are smooth points. Then we give an algorithm for constructing a set of minimal generators of the localized ideal at these points, for an arbitrary number of leaves n. This leads to a major improvement of phylogenetic reconstruction methods based on algebraic geometry.