Algebraic Geometry Seminar
Jack Huizenga
Harvard
Multiplication on P1, vector bundles, Hilbert schemes of points, and the golden ratio
Abstract: 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 $P^2$.
Wednesday March 2, 2011 at 4:00 PM in SEO 1227