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An Algorithm for Fat Points on ${{\mathbf{P}}^{2}}$

Published online by Cambridge University Press:  20 November 2018

Brian Harbourne
Brian Harbourne*
Affiliation:
Department of Mathematics and Statistics, University of Nebraska-Lincoln, Lincoln, NE 68588-0323, USA email: bharbour@math.unl.edu website: http://www.math.unl.edu/~bharbour/
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Abstract

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Let $F$ be a divisor on the blow-up $X$ of ${{\mathbf{P}}^{2}}$ at $r$ general points ${{p}_{1}},...,{{p}_{r}}$ and let $L$ be the total transform of a line on ${{\mathbf{P}}^{2}}$. An approach is presented for reducing the computation of the dimension of the cokernel of the natural map ${{\mu }_{F}}:\Gamma ({{\mathcal{O}}_{_{X}}}(F))\otimes \Gamma ({{\mathcal{O}}_{_{X}}}(L))\to \Gamma ({{\mathcal{O}}_{_{X}}}(F)\otimes {{\mathcal{O}}_{_{X}}}(L))$ to the case that $F$ is ample. As an application, a formula for the dimension of the cokernel of ${{\mu }_{_{F}}}$ is obtained when $r\,=\,7$, completely solving the problem of determining the modules in minimal free resolutions of fat point subschemes ${{m}_{1}}\,{{p}_{1}}\,+\,\cdot \cdot \cdot \,+\,{{m}_{7}}\,{{p}_{7}}\,\subset \,{{\mathbf{P}}^{2}}$. All results hold for an arbitrary algebraically closed ground field $k$.

Keywords

13P1014C9913D0213H15Generatorssyzygiesresolutionfat pointsmaximal rankplaneWeyl group
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 52 , Issue 1 , 01 February 2000 , pp. 123 - 140
Copyright
Copyright © Canadian Mathematical Society 2000

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