algebra methods for the Minimal Resolution Conjecture

If r ≥ 6, r 6= 9, we show that the Minimal Resolution Conjecture fails for a general set of γ points in P for almost 12 √ r values of γ. This strengthens the result of Eisenbud and Popescu [1999], who found a unique such γ for each r in the given range. Our proof begins like a variation of that of Eisenbud and Popescu, but uses exterior algebra methods as explained by Eisenbud and Schreyer [2000] to avoid the degeneration arguments that were the most difficult part of the EisenbudPopescu proof. Analogous techniques show that the Minimal Resolution Conjecture fails for linearly normal curves of degree d and genus g when d ≥ 3g − 2, g ≥ 4, reproving results of Schreyer, Green, and Lazarsfeld.