Let C be a Brill–Noether–Petri curve of genus g &gt;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 I_C be the ideal sheaf of a non-hyperelliptic, genus g , canonical curve. The first conjecture states that if g &gt; 8 and if the Clifford index of C is greater than 2, then H^1(^P(g−1), I^2_C(2k))=0 for k&gt;3. We prove this conjecture for g&gt;11. The second conjecture states that a Brill–Noether–Petri curve of genus g&gt;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.

Arbarello, E., Bruno, A., & Sernesi, E. (2017). On hyperplane sections of K3 surfaces. ALGEBRAIC GEOMETRY, 4(5), 562-596 [10.14231/AG-2017-028].

### On hyperplane sections of K3 surfaces

#### Abstract

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 I_C 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 H^1(^P(g−1), I^2_C(2k))=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.
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Arbarello, E., Bruno, A., & Sernesi, E. (2017). On hyperplane sections of K3 surfaces. ALGEBRAIC GEOMETRY, 4(5), 562-596 [10.14231/AG-2017-028].
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/11590/310923`
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