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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Computing the equations of a variety

Authors: Michela Brundu and Mike Stillman
Journal: Trans. Amer. Math. Soc. 337 (1993), 677-690
MSC: Primary 13P10; Secondary 13A30, 13D45, 14B15
MathSciNet review: 1091704
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Abstract: Let $ X \subset {\mathbb{P}^n}$ be a projective variety or subscheme, and let $ \mathcal{F}$ be an invertible sheaf on $ X$. A set of global sections of $ \mathcal{F}$ determines a map from a Zariski open subset of $ X$ to $ {\mathbb{P}^r}$. The purpose of this paper is to find, given $ X$ and $ \mathcal{F}$, the homogeneous ideal defining the image in $ {\mathbb{P}^r}$ of this rational map. We present algorithms to compute the ideal of the image. These algorithms can be implemented using only the computation of Gröbner bases and syzygies, and they have been implemented in our computer algebra system Macaulay. Our methods generalize to include the case when $ X$ is an arbitrary projective scheme and $ \mathcal{F}$ is generically invertible.

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Keywords: Line bundle, invertible sheaf, rational map, computing, Gröbner bases, symmetric algebra, local cohomology
Article copyright: © Copyright 1993 American Mathematical Society