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4:00 pm Tuesday, April 19, 2011 Algebraic Geometry Seminar: Multiplication on P^1, vector bundles, Hilbert schemes of points, and the golden ratioby
Jack Huizenga (Harvard) in HB 227- Consider the following basic problem about multiplication of polynomials in one variable. Fix a general 3-dimensional subspace V of the polynomials of degree a, and fix a second degree b. Given a subspace W of the polynomials of degree b, think of W as occupying the fraction dim(W)/(b+1) of the space of polynomials of degree b. For every such subspace W, does the product VW occupy at least as large a fraction of the polynomials of degree a+b as W does of the polynomials of degree b? That is, does multiplication by V always increase the fraction of the space occupied by W? Surprisingly, the answer to this question is connected to the golden ratio and its continued fraction expansion. We will further discuss how this question is connected with semistability and splitting properties of certain particularly nice vector bundles on P^2, known as Steiner bundles. These bundles can be viewed as natural generalizations of the tangent bundle. Finally, we will discuss how these bundles give rise to extremal effective divisors on the Hilbert scheme of points in P^2.
Submitted by evanmb@rice.edu |