Duke Mathematical Journal

Exterior algebra methods for the minimal resolution conjecture

David Eisenbud, Sorin Popescu, Frank-Olaf Schreyer, and Charles Walter

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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.

Article information

Duke Math. J., Volume 112, Number 2 (2002), 379-395.

First available in Project Euclid: 18 June 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13D02: Syzygies, resolutions, complexes
Secondary: 14M05: Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) [See also 13F15, 13F45, 13H10] 15A75: Exterior algebra, Grassmann algebras


Eisenbud, David; Popescu, Sorin; Schreyer, Frank-Olaf; Walter, Charles. Exterior algebra methods for the minimal resolution conjecture. Duke Math. J. 112 (2002), no. 2, 379--395. doi:10.1215/S0012-9074-02-11226-5. https://projecteuclid.org/euclid.dmj/1087575156

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