Book contents
- Frontmatter
- Contents
- List of contributors
- Preface
- 1 The effect of points fattening in dimension three
- 2 Some remarks on surface moduli and determinants
- 3 Valuation spaces and multiplier ideals on singular varieties
- 4 Line arrangements modeling curves of high degree: Equations, syzygies, and secants
- 5 Rationally connected manifolds and semipositivity of the Ricci curvature
- 6 Subcanonical graded rings which are not Cohen–Macaulay
- 7 Threefold divisorial contractions to singularities of cE type
- 8 Special prime Fano fourfolds of degree 10 and index 2
- 9 Configuration spaces of complex and real spheres
- 10 Twenty points in ℙ3
- 11 The Betti table of a high-degree curve is asymptotically pure
- 12 Partial positivity: Geometry and cohomology of q-ample line bundles
- 13 Generic vanishing fails for singular varieties and in characteristic p > 0
- 14 Deformations of elliptic Calabi–Yau manifolds
- 15 Derived equivalence and non-vanishing loci II
- 16 The automorphism groups of Enriques surfaces covered by symmetric quartic surfaces
- 17 Lower-order asymptotics for Szegö and Toeplitz kernels under Hamiltonian circle actions
- 18 Gaussian maps and generic vanishing I: Subvarieties of abelian varieties
- 19 Torsion points on cohomology support loci: From D-modules to Simpson's theorem
- 20 Rational equivalence of 0-cycles on K3 surfaces and conjectures of Huybrechts and O'Grady
- References
11 - The Betti table of a high-degree curve is asymptotically pure
Published online by Cambridge University Press: 05 January 2015
- Frontmatter
- Contents
- List of contributors
- Preface
- 1 The effect of points fattening in dimension three
- 2 Some remarks on surface moduli and determinants
- 3 Valuation spaces and multiplier ideals on singular varieties
- 4 Line arrangements modeling curves of high degree: Equations, syzygies, and secants
- 5 Rationally connected manifolds and semipositivity of the Ricci curvature
- 6 Subcanonical graded rings which are not Cohen–Macaulay
- 7 Threefold divisorial contractions to singularities of cE type
- 8 Special prime Fano fourfolds of degree 10 and index 2
- 9 Configuration spaces of complex and real spheres
- 10 Twenty points in ℙ3
- 11 The Betti table of a high-degree curve is asymptotically pure
- 12 Partial positivity: Geometry and cohomology of q-ample line bundles
- 13 Generic vanishing fails for singular varieties and in characteristic p > 0
- 14 Deformations of elliptic Calabi–Yau manifolds
- 15 Derived equivalence and non-vanishing loci II
- 16 The automorphism groups of Enriques surfaces covered by symmetric quartic surfaces
- 17 Lower-order asymptotics for Szegö and Toeplitz kernels under Hamiltonian circle actions
- 18 Gaussian maps and generic vanishing I: Subvarieties of abelian varieties
- 19 Torsion points on cohomology support loci: From D-modules to Simpson's theorem
- 20 Rational equivalence of 0-cycles on K3 surfaces and conjectures of Huybrechts and O'Grady
- References
Summary
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.
- Type
- Chapter
- Information
- Recent Advances in Algebraic GeometryA Volume in Honor of Rob Lazarsfeld’s 60th Birthday, pp. 200 - 206Publisher: Cambridge University PressPrint publication year: 2015
References
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