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


Published:20000201
Printed: Feb 2000
Abstract
Let $F$ be a divisor on the blowup $X$ 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]
