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Brill-Noether-Petri without degenerations. (English) Zbl 0608.14026

Petri’s conjecture was proved by D. Gieseker in Invent. Math. 66, 251–275 (1982; Zbl 0522.14015) by using a degeneration argument due to Griffiths and Harris. In the present paper it is proved that the general member of a complete linear system \(| C|\) on a K3 surface satisfies Petri’s condition provided every member in \(| C|\) is reduced and irreducible. This statement easily implies Petri’s conjecture. Its proof does not use degenerations; it is based on interpreting ”Petri’s condition” as a smoothness condition for a suitable map between manifolds.

MSC:

14C22 Picard groups
14J25 Special surfaces
14J28 \(K3\) surfaces and Enriques surfaces

Citations:

Zbl 0522.14015
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