Gale duality and free resolutions of ideals of points

What is the shape of the free resolution of the ideal of a general set of points in P? This question is central to the programme of connecting the geometry of point sets in projective space with the structure of the free resolutions of their ideals. There is a lower bound for the resolution computable from the (known) Hilbert function, and it seemed natural to conjecture that this lower bound would be achieved. This is the ``Minimal Resolution Conjecture'' (Lorenzini [1987], [1993]). Although the conjecture has been shown to hold in many cases, three examples discovered computationally by Frank-Olaf Schreyer in 1993 strongly suggested that it would fail in general. In this paper we shall describe a novel structure inside the free resolution of a set of points which accounts for the failure and provides a counterexample in P for every r 6; r 6ˆ 9. We begin by reviewing the conjecture and its status. Consider a set of c points in the projective r-space over a ®eld k, say C Pk. Let S ˆ k‰x0; . . . ; xrŠ, let IC be the homogeneous ideal of C, and let SC denote the homogeneous coordinate ring of C. Let