Facets of the r-stable n,k-hypersimplex

Let $k, n$ and $r$ be positive integers with $k < n$ and $r\leq\lfloor\frac{n}{k}\rfloor$. We determine the facets of the $r$-stable $n,k$-hypersimplex. As a result, it turns out that the $r$-stable $n,k$-hypersimplex has exactly $2n$ facets for every $r<\lfloor\frac{n}{k}\rfloor$. We then utilize the equations of the facets to study when the $r$-stable hypersimplex is Gorenstein. For every $k>0$ we identify an infinite collection of Gorenstein $r$-stable hypersimplices, consequently expanding the collection of $r$-stable hypersimplices known to have unimodal Ehrhart $\delta$-vectors.Comment: 12 pages, 2 figure

Normal and Δ-Normal Configurations in Toric Algebra

Toric algebra is a field of study that lies at the intersection of algebra, geometry, and combinatorics. Thus, the algebraic properties of the toric ideal IA defined by the vector configuration A are often characterizable via the geometric and combinatorial properties of its corresponding toric variety and A, respectively. Here, we focus on the property of normality of A. A normal vector configuration defines the toric ideal of a normal toric variety. However, the definition of normality of A is based entirely on the algebraic structures associated with A without regard to any of its combinatorial properties. In this paper, we discuss two attempts to provide a combinatorial characterization of normality of A. Particularly, we show that the properties the convex hull of A possesses a unimodular covering and A is a Δ-normal configuration are both sufficient conditions for normality of A, but neither is necessary. This suggests that another combinatorial property is required to provide the desired characterization of normality of A