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Gale duality and free resolutions of ideals of points

By David Eisenbud and Sorin Popescu


What is the shape of the free resolution of the ideal of a general set of points in P r? 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 r for every r 6; r 6 ˆ 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 P r k. Let S ˆ k‰x0;...; xrŠ, let IC be the homogeneous ideal of C, and let SC denote the homogeneous coordinate ring of C. Le

Year: 1999
DOI identifier: 10.1007/s002220050315
OAI identifier: oai:CiteSeerX.psu:
Provided by: CiteSeerX
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