Exterior algebra methods for the minimal resolution conjecture

Abstract

If $r\geq 6,r\neq 9$, we show that the minimal resolution conjecture (MRC) fails for a general set of $\gamma$ points in $\mathbb {P}\sp r$ for almost $(1/2)\sqrt {r}$ values of $\gamma$. This strengthens the result of D. Eisenbud and S. Popescu [EP1], who found a unique such $\gamma$ 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, G. Fløystad, and F.- O. Schreyer [EFS] to avoid the degeneration arguments that were the most difficult part of the Eisenbud-Popescu proof. Analogous techniques show that the MRC fails for linearly normal curves of degree $d$ and genus $g$ when $d\geq 3g-2,g\geq 4$, re-proving results of Schreyer, M. Green, and R. Lazarsfeld.