Phylogenetic Invariants for ℤ_3 Scheme-Theoretically

We study phylogenetic invariants of general group-based models of evolution with group of symmetries ℤ_3 . We prove that complex projective schemes corresponding to the ideal I of phylogenetic invariants of such a model and to its subideal I' generated by elements of degree at most 3 are the same. This is motivated by a conjecture of Sturmfels and Sullivant [14, Conj. 29], which would imply that I = I' .