相关论文: Vertices of Intersection Polytopes and Rays of Gen…

A Grasstope is the image of the totally nonnegative Grassmannian $\text{Gr}_{\geq 0}(k,n)$ under a linear map $\text{Gr}(k,n)\dashrightarrow \text{Gr}(k,k+m)$. This is a generalization of the amplituhedron, a geometric object of great…

The arithmetic rank of an ideal in a polynomial ring over an algebraically closed field is the smallest number of equations needed to define its vanishing locus set-theoretically. We determine the arithmetic rank of the generic $m$-residual…

We define certain natural finite sums of $n$'th roots of unity, called $G_P(n)$, that are associated to each convex integer polytope $P$, and which generalize the classical $1$-dimensional Gauss sum $G(n)$ defined over $\mathbb Z/ {n…

A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least $2k$ vertices is \textit{$k$-linked} if,…

Let $\mathscr{L} \rightarrow X$ be an ample line bundle over a complex normal projective variety $X$. We construct a flag $X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n=X$ of subvarieties for which the associated Okounkov body for…

Given a graph G, we construct a convex polytope whose face poset is based on marked subgraphs of G. Dubbed the graph multiplihedron, we provide a realization using integer coordinates. Not only does this yield a natural generalization of…

Let $(R,\mathfrak m, \mathsf k)$ be a complete intersection local ring, $K$ be the Koszul complex on a minimal set of generators of $\mathfrak m$, and $A=H(K)$ be its homology algebra. We establish exact sequences involving direct sums of…

We derive tight expressions for the maximum number of $k$-faces, $0\le k\le d-1$, of the Minkowski sum, $P_1+P_2+P_3$, of three $d$-dimensional convex polytopes $P_1$, $P_2$ and $P_3$, as a function of the number of vertices of the…

We prove that a complete intersection of $c$ very general hypersurfaces of degree at least two in $N$-dimensional complex projective space is not ruled (and therefore not rational) provided that the sum of the degrees of the hypersurfaces…

We show that the multivariate generating function of appropriately normalized moments of a measure with homogeneous polynomial density supported on a compact polytope P in R^d is a rational function. Its denominator is the product of linear…

In this article, we look at real split semisimple algebraic groups $\mathsf{G}$ with trivial center and faithful irreducible algebraic representations $\mathtt{R}$ of $\mathsf{G}$ on some vector space $\mathsf{V}$ which admit zero as a…

We show that for polytopes P_1, P_2, ..., P_r \subset \R^d, each having n_i \ge d+1 vertices, the Minkowski sum P_1 + P_2 + ... + P_r cannot achieve the maximum of \prod_i n_i vertices if r \ge d. This complements a recent result of Fukuda…

For a given convex body K in $R^d$, a random polytope $K^{(n)}$ is defined (essentially) as the intersection of $n$ independent closed halfspaces containing $K$ and having an isotropic and (in a specified sense) uniform distribution. We…

Gushel-Mukai varieties are smooth complex dimensionally transverse intersections of a cone over the Grassmannian $\mathsf{Gr}(2,5)$ with a linear space and a quadratic hypersurface. The aim of this survey is to discuss the geometry, moduli,…

Let $k$ be a number field. We refine a construction of Mestre--Shioda to construct (infinite) families of hyperelliptic curves $X/{k}$ having a record number of rational points and record Mordell--Weil rank relative to the genus of $g$ of…

Let V be a finite set of points in Euclidean d-space (d >= 2). The intersection of all unit balls B(v,1) centered at v, where v ranges over V, henceforth denoted by B(V) is the ball polytope associated with V. Note that B(V) is non-empty…

In the present paper Mori extremal rays of a smooth projective manifold X are divided into two classes: L-supported and L-negligible (where ``L'' stands for ``Lefschetz'' since the division comes from Hard Lefschetz Theorem). Roughly…

Each convex combination of extreme points of a compact convex set represents a certain point of the convex set. Barycentric coordinates provide solutions to the inverse problem of expressing an element of a compact convex set as a convex…

Let $F/\QQ $ be a totally real number field of degree $n$. We explicitly evaluate a certain sum of rational functions over a infinite fan of $F$-rational polyhedral cones in terms of the norm map $\Norm \colon F\to \QQ $. This completes…

Let $\mathbb{K}$ be a field of characteristic zero and $B=B_0+B_1$ a finite dimensional associative superalgebra. In this paper we investigate the polynomial identities of the relatively free algebras of finite rank of the variety…