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Search: MSC category 13P10 ( Grobner bases; other bases for ideals and modules (e.g., Janet and border bases) )

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1. CJM 2009 (vol 61 pp. 930)

Sidman, Jessica; Sullivant, Seth
Prolongations and Computational Algebra
We explore the geometric notion of prolongations in the setting of computational algebra, extending results of Landsberg and Manivel which relate prolongations to equations for secant varieties. We also develop methods for computing prolongations that are combinatorial in nature. As an application, we use prolongations to derive a new family of secant equations for the binary symmetric model in phylogenetics.

Categories:13P10, 14M99

2. CJM 2000 (vol 52 pp. 123)

Harbourne, Brian
An Algorithm for Fat Points on $\mathbf{P}^2
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
Categories:13P10, 14C99, 13D02, 13H15

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