On the postulation of 0-dimensional subschemes on a smooth quadric

Abstract

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

ACM sets of points in multiprojective space

“…In this situation there are several classifications. Giuffrida, Maggioni, and Ragusa [7], who helped to initiate the study of points in multiprojective spaces (see, for example [8,9,12,13,14,18,20,21,22] for more on these points), provided the first classification. They showed that ACM sets of points in P 1 × P 1 can be classified via their Hilbert functions.…”

The minimal resolutions of double points in P1×P1 with ACM support

“…Currently, only ACM sets of (fat) points in P 1 × P 1 have been classified. ACM sets of points in P 1 × P 1 were first classified via their Hilbert function in [9]. An alternative classification was provided by the second author [22], which we recall here.…”

“…Because λ = (6, 5, 3, 1, 1), it follows that α Y = (12,11,9,7,7,6,5,3,1,1). Then the shifts in the bigraded minimal free resolution of I Y are given by SY 0 = {(10, 0), (8, 1), (7, 3), (6,5), (5,6), (3,7), (2,9), (1,11), (0, 12)} SY 1 = {(10, 1), (8, 3), (7,5), (6,6), (5,7), (3,9), (2,11), (1, 12)}.…”

The regularity of points in multi-projective spaces

“…Recently, many authors (cf. [4,[9][10][11]16]) have been interested in extending our understanding of points in P n to sets of points in P n1 × · · · × P n k . We continue this trend by studying reg(I ) when I deÿnes a scheme of fat points in P n1 × · · · × P n k .…”