2015/07/17 by Enrico Arbarello, Arbarello, Enrico, Andrea Bruno +3
Mathematics · #14H51 #14J28 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #math.AG #msc:14H51 #msc:14J28
paper · pdf · doi:10.48550/arxiv.1507.05002
Title, abstract, and introduction changed (previous title: "On two conjectures by J. Wahl"). Several typos corrected. Exposition improved in various instances, according to referee's suggestions. The paper will appear in "Algebraic Geometry"
openalex publication_date 2015/07/17 · arxiv created 2016/11/13 · arxiv updated 2016/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let C be a Brill-Noether-Petri curve of genus g≥ 12. We prove that C lies on a polarized K3 surface, or on a limit thereof, if and only if the Gauss-Wahl map for C is not surjective. The proof is obtained by studying the validity of two conjectures by J. Wahl. Let IC be the ideal sheaf of a non-hyperelliptic, genus g, canonical curve. The first conjecture states that, if g≥ 8, and if the Clifford index of C is greater than 2, then H1(Pg-1, IC2(k))=0, for k≥ 3. We prove this conjecture for g≥ 11. The second conjecture states that a Brill-Noether-Petri curve of genus g≥ 12 is extendable if and only if C lies on a K3 surface. As observed in the Introduction, the correct version of this conjecture should admit limits of polarised K3 surfaces in its statement. This is what we prove in the present work.