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An Algorithm for Fat Points on
${{\mathbf{P}}^{2}}$
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- Journal:
- Canadian Journal of Mathematics / Volume 52 / Issue 1 / 01 February 2000
- Published online by Cambridge University Press:
- 20 November 2018, pp. 123-140
- Print publication:
- 01 February 2000
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- Article
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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 
$k$.
