  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.
