zbMATH — the first resource for mathematics

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

14F99 (Co)homology theory in algebraic geometry
13D99 Homological methods in commutative ring theory
14M10 Complete intersections
Full Text: Numdam EuDML
[1] Arbarello, E. , Cornalba, M. , Griffiths, P. and Harris, J. : Geometry of Algebraic Curves , v. I, Springer-Verlag, New York (1985). · Zbl 0559.14017
[2] Ballico, E. : Generators for the homogeneous ideal of s general points in P3 , J. Alg. 106 (1987) 46-52. · Zbl 0616.14043
[3] Bănică, C. and Stănăsilă, O. : Méthodes algébrigues dans la théorie globale des espaces complexes , Gauthiers-Villars, Paris (1974). · Zbl 0349.32006
[4] Fujita, T. : Defining equations for certain types of polarized varieties , in Complex Analysis and Algebraic Geometry , pp. 165-173, Cambridge University Press (1977). · Zbl 0353.14011
[5] Geramita, A. , Gregory, D. and Roberts, L. , Monomial ideals and points in projective space , J. Pure Appl. Alg. 40 (1986) 33-62. · Zbl 0586.13015
[6] Green, M. : Koszul cohomology and the geometry of projective varieties , J. Diff. Geom. 19 (1984) 125-171. · Zbl 0559.14008
[7] Green, M. and Lazarsfeld, R. : On the projective normality of complete linear series on an algebraic curve , Invent. Math. 83 (1986) 73-90. · Zbl 0594.14010
[8] Green, M. and Lazarsfeld, R. : A simple proof of Petri’s theorem on canonical curves , in Geometry Today - Giornate di geometria roma , 1984, pp. 129-142, Birkhauser, Boston (1985). · Zbl 0577.14018
[9] Homma, M. : Theorem of Enriques-Petri type for a very ample invertible sheaf on a curve of genus 3 , Math. Zeit. 183 (1983) 343-353. · Zbl 0509.14030
[10] Laksov, D. : Indecomposability of restricted tangent bundles , Astérisque 87-88 (1981) 207-220. · Zbl 0489.14008
[11] Mumford, D. : Varieties defined by quadratic equations, Corso C.I.M.E. 1969 , in Questions on Algebraic Varieties , Rome, Cremonese, pp. 30-100 (1970). · Zbl 0198.25801
[12] Mumford, D. : Lectures on curves on an algebraic surface , Ann Math. Stud. 59 (1966) · Zbl 0187.42702
[13] Serrano-Garcia, F. : A note on quadrics through an algebraic curve , preprint. · Zbl 0674.14021
[14] Saint-Donat, B. : Sur les équations définisant une courbe algebrique , C.R. Acad. Sci. Paris, Ser. A 274 (1972) 324-327. · Zbl 0234.14012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.