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^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]). Hirschowitz and Simpson [1994] showed that the conjecture holds when the number of points is large compared with r, but three examples (with r = 6,7,8) discovered computationally by Schreyer in 1993 show that the conjecture fails in general. We describe a novel structure inside the free resolution of a set of points which accounts for the observed failures and provides a counterexample in P^r for every r\geq 6, r\neq 9. The geometry behind our construction occurs not in P^r but in a different projective space, in which there is a related set of points, the ``Gale transform'' (or ``associated set'', in the sense of Coble.) 
David Eisenbud and Sorin Popescu