If X is a O-dimensional subscheme of a smooth quadric Q = P 1 x 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 Hubert function of X we study numerical properties of this matrix and their connection with the geometry of X. Further we relate 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 O-dimensional subschemes of Q.Introduction. In the last few years the interest about O-dimensional subschemes of P w has greatly grown, so many recent papers concern a deep investigation into the Hubert function, free resolution, Betti numbers, and defining equations for such subschemes. On the other hand there has been a good deal of work on two codimensional subschemes of P w hence, points of P 2 , which have both conditions, have been intensively studied. The interest on points of P 2 comes, also, because geometric properties of a variety can sometimes be given in terms of its generic hyperplane section; so, for studying curves of P 3 , one needs properties of O-dimensional subschemes of P 2 . A complete list of papers on these topics seems impossible to do; so we insert in the references just a few of them, which are more familiar to us.It seems natural to generalize this situation from one side studying O-dimensional subschemes of any variety and in particular of surfaces, on the other side working on sections of varieties done by hypersurfaces of degree bigger than one. Therefore, a first step in this direction is to investigate O-dimensional subschemes of a quadric (P 1 x P 1 ) with special regard to their behaviour with respect to the divisors of the quadric itself.When one embeds the quadric Q in P 3 , any subscheme X of Q becomes a subscheme of P 3 in that case one can relate properties of X as a subscheme of Q with those as a subscheme of P 3 . Of course, studying subschemes of Q, the geometry of the surface Q plays a big role; in particular, the cohomology groups of Q play an

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Summary. A characteristic condition is given on a zero-dimensional differentiable 0-sequence H={hi}i_>o, h~<3, in order to be the Hilbert function of a generic plane section of a reduced irreducible curve of ~,3, hence of points of ~,2 with the uniform position property. In this way an answer is given to some question stated by Harris in [Ha2].The result is obtained by constructing a smooth irreducible arithmetically Cohen-Macaulay curve in ~E)3 whose generic plane section has an assigned Hilbert function satisfying that condition.

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