Eisenbud, David; Koh, Jee-Heub Remarks on points in a projective space. (English) Zbl 0736.14022 Commutative algebra, Proc. Microprogram, Berkeley/CA (USA) 1989, Publ., Math. Sci. Res. Inst. 15, 157-172 (1989). [For the entire collection see Zbl 0721.00007.]Let \(K\) be an algebraically closed field, \(X=\{P_ 1,\ldots,P_{2r}\}\) a set of \(2r\) points in \(\mathbb{P}^ r_ K\) such that no \(2k+1\) of them lie in a \(k\)-plane. It is shown that the homogeneous ideal of \(X\) is generated by quadrics, in the case \(r\leq 4.\) This is a special case of the Green- Lazarsfeld conjecture. Moreover, the proof is done without using the linear syzygy conjecture, which is known to imply the Green-Lazarsfeld conjecture. A corresponding result is proved for sets of \(dr\) points and forms of degree \(d\), for any \(d\) and \(r\), but only scheme-theoretically. Reviewer: C.-P. Ionescu (Bucureşti) Cited in 1 ReviewCited in 17 Documents MSC: 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) 14M10 Complete intersections 14E15 Global theory and resolution of singularities (algebro-geometric aspects) Keywords:resolutions; homogeneous ideal of point; Green-Lazarsfeld conjecture Citations:Zbl 0721.00007 PDFBibTeX XMLCite \textit{D. Eisenbud} and \textit{J.-H. Koh}, in: Representations, resolutions and intertwining numbers. . 157--172 (1989; Zbl 0736.14022)