Mixtures of group-based Markov models of evolution correspond to joins of toric varieties. In this paper, we establish a large number of cases for which these phylogenetic join varieties realize their expected dimension, meaning that they are nondefective. Nondefectiveness is not only interesting from a geometric point-of-view, but has been used to establish combinatorial identifiability for several classes of phylogenetic mixture models. Our focus is on group-based models where the equivalence classes of identified parameters are orbits of a subgroup of the automorphism group of the abelian group defining the model. In particular, we show that for these group-based models, the variety corresponding to the mixture of r trees with n leaves is nondefective when n≥ 2 r+ 5. We also give improved bounds for claw trees and give computational evidence that 2-tree and 3-tree mixtures are nondefective for small n.
Bibliographical noteFunding Information:
This work began at the 2016 AMS Mathematics Research Community on “Algebraic Statistics,” which was supported by the National Science Foundation under Grant number DMS-1321794. RD was supported by NSF DMS-1401591. EG was supported by NSF DMS-1620109. RW was supported by a NSF GRF under Grant number PGF-031543, NSF RTG Grant 0943832, and a Ford Foundation Dissertation Fellowship. HB was supported in part by a research assistantship, funded by the National Institutes of Health Grant R01 GM117590. PEH was partially supported by NSF Grant DMS-1620202.