Related papers: A note on nontrivial intersection for selfmaps of …

We investigate the existence of closed polylines (also known as closed polygonal chains or self-crossing polygons) that intersect each of their edges the same number of times. The most general question in this corner of combinatorial…

Let $V$ be an $(n+\ell)$-dimensional vector space over a finite field, and $W$ a fixed $\ell$-dimensional subspace of $V$. Write ${V\brack n,0}$ to be the set of all $n$-dimensional subspaces $U$ of $V$ satisfying $\dim(U\cap W)=0$. A…

The Grassmannians of lines in projective N-space, G(1,N), are embedded by way of the Pl"ucker embedding in the projective space P(\bigwedge^2 C^{N+1}). Let H^l be a general l-codimensional linear subspace in this projective space. We…

Let $G$ be a cyclic $p$-group or generalized quaternion group, $X\in \pi_0 S_G$ be a virtual $G$-set, and $V$ be a fixed point free complex $G$-representation. Under conditions depending on the sizes of $G$, $X$, and $V$, we construct a…

Grassmann manifolds $G_{k,n}$ are among the central objects in geometry and topology. The Borel picture of the mod 2 cohomology of $G_{k,n}$ is given as a polynomial algebra modulo a certain ideal $I_{k,n}$. The purpose of this paper is to…

Consider a face F in an arrangement of n Jordan curves in the plane, no two of which intersect more than s times. We prove that the combinatorial complexity of F is O(\lambda_s(n)), O(\lambda_{s+1}(n)), and O(\lambda_{s+2}(n)), when the…

A graph on $2k+1$ vertices consisting of $k$ triangles which intersect in exactly one common vertex is called a $k$-fan and denoted by $F_k$. This paper aims to determine the graphs of order $n$ that have the maximum (adjacency) spectral…

Let G(d,n) denote the Grassmannian of d-planes in C^n and let T be the torus (C^*)^n/diag(C^*) which acts on G(d,n). Let x be a point of G(d,n) and let \bar{Tx} be the closure of the T-orbit through x. Then the class of the structure sheaf…

We show that when $d \geq 3$ and $m>2$, the Nef cone of the Hilbert scheme $Hilb_{P_{d,m}(T)}(G(k,n))$ is a cone spanned by 6 classes in general case, where $P_{d,m}(T)=\binom{T+m}{m}-\binom{T+m-d}{m}$.

Let $Gr(k,n)$ be the Pl\"ucker embedding of the Grassmann variety of projective $k$-planes in $\P n$. For a projective variety $X$, let $\sigma_s(X)$ denote the variety of its $s-1$ secant planes. More precisely, $\sigma_s(X)$ denotes the…

If f maps a discrete d-manifold G onto a (k+1)-partite complex P then H(G,f,P),the set of simplices x in G such that f(x) contains at least one facet in P defines a (d-k)-manifold.

Let $[n]^{(k)}$ be the set of all ordered $k$-tuples of distinct elements in $[n]=\{1,2,...,n\}$. The $(n,k,r)$-arrangement graph $A(n,k,r)$ with $1\leq r\leq k\leq n$, is the graph with vertex set $[n]^{(k)}$ and with two $k$-tuples are…

We show that the Gromov width of the Grassmannian of complex k-planes in C^n is equal to one when the symplectic form is normalized so that it generates the integral cohomology in degree 2. We deduce the lower bound from more general…

Let $\mathcal{KC}_g ^k$ be the moduli stack of pairs $(S,C)$ with $S$ a $K3$ surface and $C\subset S$ a genus $g$ curve with divisibility $k$ in $\mathrm{Pic}(S)$. In this article we study the forgetful map $c_g^k:(S,C) \mapsto C$ from…

We define a $C(k)$ to be a family of $k$ sets $F_1,\dots,F_k$ such that $\textrm{conv}(F_i\cup F_{i+1})\cap \textrm{conv}(F_j\cup F_{j+1})=\emptyset$ when $\{i,i+1\}\cap \{j,j+1\}=\emptyset$ (indices are taken modulo $k$). We show that if…

The Johnson graph J(v,k) has, as vertices, the k-subsets of a v-set V, and as edges the pairs of k-subsets with intersection of size k-1. We introduce the notion of a neighbour-transitive code in J(v,k). This is a vertex subset \Gamma such…

Given a collection of planar graphs $G_1,\dots,G_k$ on the same set $V$ of $n$ vertices, the simultaneous geometric embedding (with mapping) problem, or simply $k$-SGE, is to find a set $P$ of $n$ points in the plane and a bijection $\phi:…

In this paper, we give a full classification of all homogeneous Ulrich bundles on a Grassmannian $\Gr(k,n)$ of $k$-planes on $\PP^n$.

Let $G$ be a connected graph on $n$ vertices and $1 \le k \le n-1$ an integer. The $k$-token graph of $G$ is the graph $F_k(G)$ whose vertices are all the $k$-subsets of vertices of $G$, two of which are adjacent whenever their symmetric…

The family of the complex Grassmann manifolds $G_{n,k}$ with a canonical action of the torus $T^n=\mathbb{T}^{n}$ and the analogue of the moment map $\mu : G_{n,k}\to \Delta _{n,k}$ for the hypersimplex $\Delta _{n,k}$, is well known. In…