The Eisenbud-Koh-Stillman Conjecture on Linear Syzygies

Abstract

It is proved, as was conjectured by Eisenbud-Koh-Stillman, that for a finitely generated graded module $M$ over the symmetric algebra $S(V)$, if the Koszul group ${\mathcal{K}}_{p,0}(M,V)\ne 0$, then the set of rank 1 relations in $M_0\otimes V$ has dimension $\ge p$. The method is by using ``exterior minors" to study syzygies of an ideal derived from the Koszul class in the exterior algebra. As a consequence, a conjecture of Lazarsfeld and myself that a set $Z$ of $2r+1-p$ points in ${\mathbf{P}}^r$ for which property $N_p$ fails has a subset $Z^{\prime }$ of at least $2\dim (L)+2-p$ points lying on a linear space $L$ such that property $N_p$ fails for $Z^{\prime }$.

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