  If r\geq 6, r\neq 9, we show that the Minimal Resolution Conjecture fails for
a general set of m points in P^r for almost 1/2\sqrt r values of m. This
strengthens the result of Eisenbud and Popescu [1999], who found a unique such
m 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 Eisenbud-Popescu proof. Analogous techniques show
that the Minimal Resolution Conjecture fails for linearly normal curves of
degree d and genus g when d\geq 3g-2, g\geq 4, reproving results of Schreyer,
Green, and Lazarsfeld.
