On hyperplane sections of K3 surfaces

Enrico Arbarello, Andrea Bruno, Edoardo Sernesi

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)


Let C be a Brill-Noether-Petri curve of genus g ≥ 12. We prove that C lies on a polarised 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.

Original languageEnglish
Pages (from-to)562-596
Number of pages35
JournalAlgebraic Geometry
Issue number5
Publication statusPublished - 1 Nov 2017
Externally publishedYes


  • Brill-Noether-Petri curves
  • Gauss-Wahl map
  • K3 surfaces


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