Local equations for equivariant evolutionary models

Phylogenetic varieties related to equivariant substitution models have been studied largely in the last years. One of the main objectives has been finding a set of generators of the ideal of these varieties, but this has not yet been achieved in some cases (for example, for the general Markov model...

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Vydané v:	Advances in mathematics (New York. 1965) Ročník 315; s. 285 - 323
Hlavní autori:	Casanellas, Marta, Fernández-Sánchez, Jesús, Michałek, Mateusz
Médium:	Journal Article Publikácia
Jazyk:	English
Vydavateľské údaje:	Elsevier Inc 31.07.2017
Predmet:	14 Algebraic geometry 14D Families, fibrations 60 Probability theory and stochastic processes 60J Markov processes 92 Biology and other natural sciences 92D Genetics and population dynamics Classificació AMS Complete intersection Evolutionary model Filogènia Geometria algebraica Geometry, Algebraic Matemàtiques i estadística Phylogenetic tree Phylogenetic variety Phylogeny Representation theory Àrees temàtiques de la UPC 60J20 Phylogenetic variety Evolutionary model Complete intersection 92D15 Phylogenetic tree Representation theory 14H10
ISSN:	0001-8708, 1090-2082
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Popis
Shrnutí:	Phylogenetic varieties related to equivariant substitution models have been studied largely in the last years. One of the main objectives has been finding a set of generators of the ideal of these varieties, but this has not yet been achieved in some cases (for example, for the general Markov model this involves the open “salmon conjecture”, see [2]) and it is not clear how to use all generators in practice. Motivated by applications in biology, we tackle the problem from another point of view. The elements of the ideal that could be useful for applications in phylogenetics only need to describe the variety around certain points of no evolution (see [13]). We produce a collection of explicit equations that describe the variety on a Zariski open neighborhood of these points (see Theorem 5.4). Namely, for any tree T on any number of leaves (and any degrees at the interior nodes) and for any equivariant model on any set of states κ, we compute the codimension of the corresponding phylogenetic variety. We prove that this variety is smooth at general points of no evolution and, if a mild technical condition is satisfied (“d-claw tree hypothesis”), we provide an algorithm to produce a complete intersection that describes the variety around these points.
ISSN:	0001-8708 1090-2082
DOI:	10.1016/j.aim.2017.05.003