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4. Two existence theorems

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1842)

Abstract

Given integers \(g \geq 2\) and \(0 \leq c \leq \lfloor \frac{g-1}{2} \rfloor\), one may ask whether there actually exists a pair (S,L), where S is a K3 surface, L 2 = 2g – 2 and all smooth curves in |L| have Clifford index c.

Theorem 4.1 below gives a positive answer to this question. Theorem 4.4 below answers the same kind of question concerning the possible gonalities of a curve on a K3 surface.

The results in this chapter were first given in [Kn3]. We also include the material here, to obtain a complete exposition.

Mathematics Subject Classification (2000):

  • 14J28
  • 14H51

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Correspondence to Trygve Johnsen .

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© 2004 Springer-Verlag Berlin/Heidelberg

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Johnsen, T., Knutsen, A.L. (2004). 4. Two existence theorems. In: K3 Projective Models in Scrolls. Lecture Notes in Mathematics, vol 1842. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-40898-7_4

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  • DOI: https://doi.org/10.1007/978-3-540-40898-7_4

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-21505-9

  • Online ISBN: 978-3-540-40898-7

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