Dedicated to Rob Lazarsfeld on the occasion of his 60th birthday

1 Introduction

Syzygies can encode subtle geometric information about an algebraic variety, with the most famous examples coming from the study of smooth algebraic curves. Though little is known about the syzygies of higher-dimensional varieties, Ein and Lazarsfeld have shown that at least the asymptotic behavior is uniform [1]. More precisely, given a projective variety X ⊆ ℙn embedded by the very ample bundle A, Ein and Lazarsfeld ask: which graded Betti numbers are nonzero for X re-embedded by dA? They prove that, asymptotically in d, the answer (or at least the main term of the answer) only depends on the dimension of X.

Boij-Söderberg theory [4] provides refined invariants of a graded Betti table, and it is natural to ask about the asymptotic behavior of these Boij-Söderberg decompositions. In fact, this problem is explicitly posed by Ein and Lazarsfeld [1, Problem 7.4], and we answer their question for smooth curves in Theorem 3.

Fix a smooth curve C and a sequence {Ad} of increasingly positive divisors on C. We show that, as d → ∞, the Boij-Söderberg decomposition of the Betti table of C embedded by |Ad| is increasingly dominated by a single pure diagram that depends only on the genus of the curve. The proof combines an explicit computation about the numerics of pure diagrams with known facts about when an embedded curve satisfies Mark Green's Np-condition.

2 Setup

We work over an arbitrary field k. Throughout, we will fix a smooth curve C of genus g and a sequence {Ad} of line bundles of increasing degree. Since we are interested in asymptotics, we assume that for all d, deg Ad ≥ 2g + 1. Let rd := dim H0(C, Ad) − 1 = deg Ad − g so that the complete linear series |Ad| embeds C ⊆ ℙrd.