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SINGULAR CURVES ON A K3 SURFACE AND LINEAR SERIES ON THEIR NORMALIZATIONS

FLAMINIO FLAMINI

Dipartimento di Matematica, Università di Roma "Tor Vergata", Via della Ricerca Scientifica, 1, 00133 Roma, Italy

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ANDREAS LEOPOLD KNUTSEN

Dipartimento di Matematica, Università "Roma Tre", Largo S. Leonardo Murialdo, 1, 00146 Roma, Italy

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GIANLUCA PACIENZA

Institut de Recherche Mathématique Avancée, Université, L. Pasteur et CNRS 7, rue R. Descartes - 67084 Strasbourg Cedex, France

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https://doi.org/10.1142/S0129167X0700428XCited by:10 (Source: Crossref)

Abstract

We study the Brill–Noether theory of the normalizations of singular, irreducible curves on a K3 surface. We introduce a singular Brill–Noether number ρ_sing and show that if Pic(K3) = ℤ[L], there are no 's on the normalizations of irreducible curves in |L|, provided that ρ_sing < 0. We give examples showing the sharpness of this result. We then focus on the case of hyperelliptic normalizations, and classify linear systems |L| containing irreducible nodal curves with hyperelliptic normalizations, for ρ_sing < 0, without any assumption on the Picard group.

Keywords:

AMSC: Primary 14H10, Secondary 14H51, Secondary 14J28, Secondary 14J60

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