Compositio Mathematica

Table of Contents - 01 May 2011 - Volume 147, Issue 03  

Research Article

Green’s conjecture for curves on arbitrary K3 surfaces

Marian Aprodua1a2 and Gavril Farkasa3

a1 Institute of Mathematics ‘Simion Stoilow’ of the Romanian Academy, RO-014700 Bucharest, Romania (email: aprodu@imar.ro)

a2 Şcoala Normală Superioară Bucureşti, Calea Griviţei 21, Sector 1, RO-010702 Bucharest, Romania

a3 Humboldt-Universität zu Berlin, Institut Für Mathematik, 10099 Berlin, Germany (email: farkas@math.hu-berlin.de)

Abstract

Green’s conjecture predicts than one can read off special linear series on an algebraic curve, by looking at the syzygies of its canonical embedding. We extend Voisin’s results on syzygies of K3 sections, to the case of K3 surfaces with arbitrary Picard lattice. This, coupled with results of Voisin and Hirschowitz–Ramanan, provides a complete solution to Green’s conjecture for smooth curves on arbitrary K3 surfaces.

(Received December 02 2009)

(Accepted June 15 2010)

(Online publication February 15 2011)

2000 Mathematics Subject Classification

  • 13D02;
  • 14C20

Keywords

  • syzygy;
  • canonical curve;
  • Brill–Noether theory