The well-known Erdős–Ko–Rado Theorem states that if F is a family of k-element subsets of {1, 2, . . . , n} (n ≥ 2k) satisfying S, T ∈ F ⇒ |S ∩ T | ≥ 1, then |F| ≤ ( n−1 k−1 ) . The theorem also provides necessary and sufficient conditions for attaining the maximum. We present elementary methods for deriving generalizations of the Erdős– Ko–Rado Theorem on several classes of combinatorial objec...

The goal of this paper is to prove Bézout’s Theorem for algebraic curves. Along the way, we introduce some basic notions in algebraic geometry such as affine and projective varieties and intersection numbers of algebraic curves. After proving Bézout’s Theorem, we investigate Max Noether’s Fundamental Theorem and construct the group law on a plane cubic which is the first step in studying the ar...

The main topic of this paper is various " hyperbolic " generalizations of the Edmonds-Rado theorem on the rank of intersection of two matroids. We prove several results in this direction and pose a few questions. We also give generalizations of the Obreschkoff theorem and recent results of J. Borcea and B. Shapiro.

We prove a decomposition theorem for the cd-index of a Gorenstein* poset analogous to the decomposition theorem for the intersection cohomology of a toric variety. From this we settle a conjecture of Stanley that the cd-index of Gorenstein* lattices is minimized on Boolean algebras.

We show that standard arguments for deformations based on dimension counts can also be applied over a (not necessarily Noetherian) valuation ring A of rank 1. Key intermediate results are a principal ideal theorem for schemes of finite type over A, and a theorem on subadditivity of intersection codimension for schemes smooth over A.

We give a new proof of Delsarte, Goethals and Mac williams theorem on minimal weight codewords of generalized Reed-Muller codes published in 1970. To prove this theorem, we consider intersection of support of minimal weight codewords with affine hyperplanes and we proceed by recursion.

The theta divisor Θ of the Jacobian variety of a complex curve X is best viewed as a divisor inside the component Pic(X) consisting of (isomorphism classes of) line bundles of degree g−1. Then a line bundle L belongs to Θ if and only if it has non-zero sections. Riemann proved that the multiplicity of Θ at a point L is equal to dimH(X;L) − 1. Kempf ([Ke]) obtained a geometric proof of Riemann’s...

In this paper, we propose a new height function for a variety defined over a finitely generated field overQ. For this height function, we will prove Northcott’s theorem and Bogomolov’s conjecture, so that we can recover the original Raynaud’s theorem (Manin-Mumford’s conjecture). CONTENTS Introduction 1 1. Arakelov intersection theory 3 2. Arithmetically positive hermitian line bundles 6 3. Ari...

In this paper, we continue the investigation of intersections of pairs of finitely generated subgroups Γ1 and Γ2 of a Kleinian group Γ, a question which has been examined by a number of authors. We give a brief survey of this work at the end of the introduction. We consider here intersections of topologically tame subgroups of Γ. We are interested primarily in determining the connection between...

The fundamental result of Edmonds [5] started the area of packing arborescences and the great number of recent results shows increasing interest of this subject. Two types of matroid constraints were added to the problem in [2, 3, 9], here we show that both contraints can be added simultaneously. This way we provide a solution to a common generalization of the reachability-based packing of arbo...