A vanishing theorem for weight-one syzygies

Abstract

Abstract. We give a criterion for the vanishing of the weight one syzygies associated to a line bundle B in a sufficiently positive embedding of a smooth complex projective variety of arbitrary dimension.

Syzygies of projective varieties of large degree: Recent progress and open problems

“…However as observed [9, §5], one can hope for more precise results for the groups K p,1 (X, B; L d ): in particular, for d 0 one can expect that the values of p for which these groups vanish to depend only on the geometry of B. Results along these lines were established in [10] and [11]. The case of curves, treated in [10], is particularly interesting as it leads to the proof of an old conjecture from [21], so we start with this.…”

“…Inspired by Yang's interpretation of Koszul cohomology in [37], Yang and the authors establish in [11] the following: Conversely, if there is a reduced zero cycle w = x 1 + . .…”

“…Section 3 centers on some results and conjectures concerning the asymptotics of Betti numbers. Finally, we return to curves in §4, and explain the proof of the gonality conjecture from [10] and discuss briefly the extension in [11] of this result to higher dimensions.…”

Asymptotic vanishing of syzygies of algebraic varieties

“…However, if H q−1 (X, B) = 0 for some 2 ≤ q ≤ n, then K r d −q+1,q (X, B; L d ) = 0 for large d (see [12,Remark 5.3]). When X is a smooth projective complex variety and B is a line bundle, vanishing of weight-one syzygies K p,1 (X, B; L d ) for large p was treated in the work of Ein-Lazarsfeld-Yang [15], which can be seen as a generalization of Green-Lazarsfeld's gonality conjecture. See [13,29] for the proof of the gonality conjecture.…”

Divisor varieties of symmetric products

“…This latter bundle in particular has enjoyed much interest in the literature, e.g. in [7] and [8]. We will write n(d) := c 1 (N L ) and t(d) := c 1 (T L ) for L of degree d.…”