Restrictions of Steiner bundles and divisors on the Hilbert scheme of points in the plane

Jack Huizenga

Published 2011-02-28, updated 2012-03-01Version 2

The Hilbert scheme of n points in the projective plane parameterizes degree n zero-dimensional subschemes of the projective plane. We examine the dual cones of effective divisors and moving curves on the Hilbert scheme. By studying interpolation, restriction, and stability properties of certain vector bundles on the plane we fully determine these cones for just over three fourths of all values of n. A general Steiner bundle on P^N is a vector bundle admitting a linear resolution of a particular form. We complete the classification of slopes of semistable Steiner bundles on P^N by showing every admissible slope is realized by a bundle which restricts to a semistable bundle on a rational curve. The proof involves a basic question about multiplication of polynomials on P^1 which is interesting in its own right.