Projections of projective curves with higher linear syzygies

Abstract

In this article, we prove that the inner projection of a projective curve with higher linear syzygies has also higher linear syzygies. Specifically, if a very ample line bundle L on a smooth projective curve X satisfies property N(p) for p >= 1 and H(1) (L(circle times)2) = 0, then 2(-q) satisfies property N(p-1) for any point q is an element of X. We also give simple proofs of well-known theorems about syzygies and raise some questions related to the line bundles of degree 2g which do not satisfy property N(1).