On the geometry of binary symmetric models of phylogenetic trees

Abstract

We investigate projective varieties which are binary symmetric models of trivalent phylogenetic trees. We prove that they have Gorenstein terminal singularities and are Fano varieties of index 4 and dimension equal to the number of edges of the tree in question. Moreover any two such varieties which are of the same dimension are deformation equivalent, that is, they are in the same connected component of the Hilbert scheme of the projective space. As an application we provide a simple formula for computing their Hilbert-Ehrhart polynomial.

Presentations of semigroup algebras of weighted trees

“…Using this procedure on any (T * , e * , ω T * ), and (T , e, ω T ) for any edge e ∈ T , can create a new weighted tree by identifying the new 0-weighted edges. On the level of the combinatorics of the trees, this construction is called the graft of two pointed trees, and was introduced in Definition 2.25 of [3]. An example is pictured in Figure 17.…”

“…The reason for this resemblance is not entirely accidental, see [5] for a moduli-of-surfaces interpretation of spaces associated to the semigroup S T . Buczynska and Wisniewski define merging in [3], where they show that a similar fibered product formula holds for a class of semigroups of weightings which we will now introduce.…”

“…It is shown in [3] (proposition 1.13) that (T ) is a fibered product of |I (T )| copies of (Y ). The lattice point semigroup of L (T ) = (T ) + .…”

“…We observe that the lattice points of (Y ) are given by the degree 1 members of S 1 Y . In [3] Buczynska and Wisniewski study the algebras C[S 1 T ], proving that they are all deformation equivalent. However, they do not construct an analogue of the projective coordinate ring of the Grassmannian of two planes in this context, namely an algebra for each n which flatly degenerates to the semigroup algebra C[S 1 T ] for each tree T with n leaves while preserving the multigrading defined by the edge weights and the level.…”

Using Invariants for Phylogenetic Tree Construction

“…On the theoretical side of phylogenetics, they have been used to answer questions about identifiability (e.g., [3,37]). The study of the algebraic geometry arising from invariants has led to many interesting problems in mathematics [18,9,15].…”