Geometry of the Kimura 3-parameter model

Details Geometry of the Kimura 3-parameter model Geometry of the Kimura 3-parameter model

2007-02-27

arxiv.org/abs/math/0702834v1

Mathematics

Algebraic Geometry

26 pages with 4 figures

Scientific paper

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.

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