The well-poised property and torus quotients

Abstract

An embedded variety is said to be well-poised when the associated initial ideal degenerations coming from points of the tropical variety are reduced and irreducible. Varieties with a well-poised embedding admit a large collection of explicitly constructible Newton-Okounkov bodies. This paper aims to study the well-poised property under torus quotients. Our first result states that GIT quotients of normal well-poised varieties by quasi-tori also have well-poised embeddings. As an application, we show that several Hassett spaces, $\overline{M}_{0,\beta}$, are well-poised under Alexeev's embedding. Conversely, given an affine $T$-variety $X$ with polyhedral divisor $\mathfrak{D}$ on a well-poised base $Y$, we construct an embedding of $X \subseteq \mathbb{A}^N$ and provide conditions on $Y$ and $\mathfrak{D}$ which if met, imply $X$ is well-poised under this embedding. Then we show that any affine arrangement variety meets the specified criteria, generalizing results of Ilten and the second author for rational complexity 1 varieties. Using this result, we explicitly compute many Newton-Okounkov cones of $X$ and provide a criterion for the associated toric degenerations to be normal. Our final application combines these two results to show that hypertoric varieties have well-poised embeddings.