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Some results on the syzygies of finite sets and algebraic curves. (English) Zbl 0671.14010
Let $$X\subseteq {\mathbb{P}}^ r$$ be a projective variety, not contained in any hyperplane, with homogeneous ideal I. Let $$E_{\bullet}$$ be a minimal graded free resolution of I over the homogeneous coordinate ring S of $${\mathbb{P}}^ r:$$ $$0\to E_{r+1}\to...\to E_ 1\to I\to 0$$, where $$E_ i=\oplus S(-a_{ij})$$. Let $$N_ p$$ be the following property: $$E_ i=\oplus S(-i-1)$$ for $$1\leq i\leq p$$. (For example, $$N_ 1$$ means that I is generated by quadrics.) Then the authors prove that if X consists of $$2r+1-p$$ points, no $$r+1$$ lying on a hyperplane, then X satisfies property $$N_ p.$$ From this they reprove an earlier result of one of the authors [M. L. Green, J. Differ. Geom. 19, 125-171 (1984; Zbl 0559.14008)]: Let X be a smooth irreducible projective curve of genus g, and for $$p\geq 1$$ consider the imbedding $$X\subseteq P^{g+p+1}$$ defined by the complete linear system associated to a line bundle of degree $$2g+1+p$$. Then X satisfies property $$(N_ p).$$ Finally the authors characterize the curves X, together with a line bundle of degree $$2g+p$$, for which $$N_ p$$ fails.
These results generalize earlier work of Mumford, Fujita, and Saint Donat.
Reviewer: Leslie G.Roberts

##### MSC:
 14F99 (Co)homology theory in algebraic geometry 13D99 Homological methods in commutative ring theory 14M10 Complete intersections
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##### References:
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