Restrictions of Steiner Bundles and Divisors on the Hilbert Scheme of Points in the Plane

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Restrictions of Steiner Bundles and Divisors on the Hilbert Scheme of Points in the Plane

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Title: Restrictions of Steiner Bundles and Divisors on the Hilbert Scheme of Points in the Plane
Author: Huizenga, Jack
Citation: Huizenga, Jack. 2012. Restrictions of Steiner Bundles and Divisors on the Hilbert Scheme of Points in the Plane. Doctoral dissertation, Harvard University.
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Abstract: 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 \(\mathbb{P}^N\) is a vector bundle \(E\) admitting a resolution of the form \(0 \rightarrow \mathcal{O}_{\mathbb{P}^N} (−1)^s {M \atop \rightarrow} \mathcal{O}^{s+r}_{\mathbb{P}^N} \rightarrow E \rightarrow 0\), where the map \(M\) is general. We complete the classification of slopes of semistable Steiner bundles on \(\mathbb{P}^N\) by showing every admissible slope is realized by a bundle which restricts to a balanced bundle on a rational curve. The proof involves a basic question about multiplication of polynomials on \(\mathbb{P}^1\) which is interesting in its own right.
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Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:9571108
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