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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.

14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14M10 Complete intersections
14E15 Global theory and resolution of singularities (algebro-geometric aspects)