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

  Published:2000-02-01
 Printed: Feb 2000
  • Brian Harbourne
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Abstract

Let $F$ be a divisor on the blow-up $X$ of $\pr^2$ at $r$ general points $p_1, \dots, p_r$ and let $L$ be the total transform of a line on $\pr^2$. An approach is presented for reducing the computation of the dimension of the cokernel of the natural map $\mu_F \colon \Gamma \bigl( \CO_X(F) \bigr) \otimes \Gamma \bigl( \CO_X(L) \bigr) \to \Gamma \bigl( \CO_X(F) \otimes \CO_X(L) \bigr)$ 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\break $m_1 p_1 + \cdots + m_7 p_7 \subset \pr^2$. All results hold for an arbitrary algebraically closed ground field~$k$.
Keywords: Generators, syzygies, resolution, fat points, maximal rank, plane, Weyl group Generators, syzygies, resolution, fat points, maximal rank, plane, Weyl group
MSC Classifications: 13P10, 14C99, 13D02, 13H15 show english descriptions Grobner bases; other bases for ideals and modules (e.g., Janet and border bases)
None of the above, but in this section
Syzygies, resolutions, complexes
Multiplicity theory and related topics [See also 14C17]
13P10 - Grobner bases; other bases for ideals and modules (e.g., Janet and border bases)
14C99 - None of the above, but in this section
13D02 - Syzygies, resolutions, complexes
13H15 - Multiplicity theory and related topics [See also 14C17]
 

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