The Eisenbud-Koh-Stillman conjecture on linear syzygies

Abstract

De®nition 0.1. A variety X and line bundle L has property Np for p 0 an integer if: (1) p ˆ 0 It is normally generated, i.e. SkH0 X ;L† ! H0 X ; L k† is surjective for all k 0; (2) p ˆ 1 It is normally generated and the kernel of k 0SH X ; L† ! k 0H X ; L k† is generated by quadrics; (3) p 2 It has properties N0 and N1 and the k'th syzygies among the quadrics are generated by linear syzygies for all 1 k p y 1.