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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.

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