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

Harbourne, Brian

Canadian journal of mathematics, Tome 52 (2000) no. 1, pp. 123-140

Résumé

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 $k$ .

Détail

DOI : 10.4153/CJM-2000-006-6

Mots-clés : 13P10, 14C99, 13D02, 13H15, Generators, syzygies, resolution, fat points, maximal rank, plane, Weyl group

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