This article follows the paper of Griffiths and Harris, "On the variety of special linear systems on a general algebraic curve." 1. WARMUP ON DEGENERATIONS The classic first problem in Schubert calculus is: how many lines intersect four general lines in P 3 ? First, what does this numerology come from? The space of lines in P 3 is G(1, 3) which has dimension 2 × 2 = 4. The locus of lines inters...

Let Bi be an (Ni, d)-fractional Brownian motion with Hurst index αi (i = 1,2), and let B1 and B2 be independent. We prove that, if N1 α1 + N2 α2 > d , then the intersection local times of B1 and B2 exist, and have a continuous version. We also establish Hölder conditions for the intersection local times and determine the Hausdorff and packing dimensions of the sets of intersection times and int...

In the early 1970s, Bro. U. Alfred Brousseau asked for the number of regions formed in an infinite strip by the mn segments that join m equally spaced points on one edge to n equally spaced points on the other. Using projective duality, we express the number of points, segments, and regions formed by Brousseau's configuration in terms of the numbers Lk(m, n) of lines that meet an m x n lattice ...

Given a set P of n points in the plane, its separability is the minimum number of lines needed to separate all its pairs of points from each other. We show that the minimum number of lines needed to separate n points, picked randomly (and uniformly) in the unit square, is Θ̃(n), where Θ̃ hides polylogarithmic factors. In addition, we provide a fast approximation algorithm for computing the separa...

I: intersection graphs of intervals on the plane; R: intersection graphs of rays on the plane; S: intersection graphs of families of strings on the plane such that the intersection of any two strings is a connected subset of the plane; Im : intersection graphs of intervals on the plane parallel to some m lines; Rm : intersection graphs of rays on the plane parallel to some m lines. In particula...

In this paper we study an incidence problem in finite geometry. Suppose we are given a set X which is the union of some number of lines in Fq . We choose a subset Y of X, such that for each line ` in X, at least half of its points are in Y . We show that |Y | is always at least some fraction of |X|. Using the polynomial method and degree reduction, it was previously known that such a statement ...