Gorenstein Semigroup Algebras of Weighted Trees

We classify exactly when the toric algebras \C[S_{\tree}(\br)] are Gorenstein. These algebras arise as toric deformations of algebras of invariants of the Cox-Nagata ring of the blow-up of $n-1$ points on $\mathbb{P}^{n-3}$, or equivalently algebras of the ring of global sections for the Pl\"ucker embedding of weight varieties of the Grassmanian Gr_2(\C^n), and algebras of global sections for embeddings of moduli of weighted points on $\mathbb{P}^1$. As a corollary, we find exactly when these families of rings are Gorenstein as well.Comment: 11 Pages, 7 Figures, expanded proof of proposition 3.

Presentations of Semigroup Algebras of Weighted Trees

We find presentations for subalgebras of invariants of the coordinate algebras of binary symmetric models of phylogenetic trees studied by Buczynska and Wisniewski in \cite{BW}. These algebras arise as toric degenerations of rings of global sections of weight varieties of the Grassmanian of two planes associated to the Pl\"ucker embedding, and as toric degenerations of rings of invariants of Cox-Nagata rings.Comment: 17 pages; 16 figures; Shortened proof of Theorems 2.2 and 2.4, changed abstrac