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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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