We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
${{\mathbf{P}}^{2}}$
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 
$k$.
