Restrictions of steiner bundles and divisors on the hilbert scheme of points in the plane

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Abstract

The Hilbert scheme of n points in the projective plane parameterizes degree n zerodimensional 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 PN is a vector bundle E admitting a resolution of the form 0→OPN (-1)s M→Os+rP N →E →0,where the map M is general. We complete the classification of slopes of semistable Steiner bundles on PN by showing every admissible slope is realized by a bundle which restricts itself to a balanced bundle on a rational curve. The proof involves a basic question about multiplication of polynomials on P1 which is interesting in its own right.

Original languageEnglish (US)
Pages (from-to)4829-4873
Number of pages45
JournalInternational Mathematics Research Notices
Volume2013
Issue number21
DOIs
StatePublished - Jan 1 2013

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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