Gorenstein Semigroup Algebras of Weighted Trees

TL;DR

This paper classifies when certain toric algebras associated with weighted trees are Gorenstein, linking algebraic properties to geometric and combinatorial structures like Grassmannians and moduli spaces.

Contribution

It provides a complete classification of Gorenstein properties for toric algebras arising from weighted trees, connecting algebraic, geometric, and combinatorial frameworks.

Findings

Identifies precise conditions for Gorenstein property in these algebras.

Connects algebraic classification to geometric structures like Grassmannians.

Provides criteria for Gorenstein property in families of related rings.

Abstract

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.