On the postulation of 0-dimensional subschemes on a smooth quadric. (English) Zbl 0723.14035

Giuffrida, Salvatore; Maggioni, Renato; Ragusa, Alfio

Pac. J. Math. 155, No. 2, 251-282 (1992).

If X is a 0-dimensional subscheme of a smooth quadric \(Q\cong {\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\) we investigate the behaviour of X with respect to the linear systems of divisors of any degree (a,b). This leads to the construction of a matrix of integers which plays the role of a Hilbert function of X; we study numerical properties of this matrix and their connection with the geometry of X. Further we put into relation the graded Betti numbers of a minimal free resolution of X on Q with that matrix, and give a complete description of the arithmetically Cohen- Macaulay 0-dimensional subschemes of Q.

Reviewer: S.Giuffrida

MSC:

14M05	Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14N05	Projective techniques in algebraic geometry
13D40	Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series

Keywords:

postulation; arithmetically Cohen-Macaulay 0-dimensional subschemes of a smooth quadric; linear systems of divisors; Hilbert function