Smooth curves on projective K3 surfaces
Abstract
In this paper we give for all $n \geq 2$, d>0, $g \geq 0$ necessary and sufficient conditions for the existence of a pair (X,C), where X is a K3 surface of degree 2n in $\matbf{P}^{n+1}$ and C is a smooth (reduced and irreducible) curve of degree d and genus g on X. The surfaces constructed have Picard group of minimal rank possible (being either 1 or 2), and in each case we specify a set of generators. For $n \geq 4$ we also determine when X can be chosen to be an intersection of quadrics (in all other cases X has to be an intersection of both quadrics and cubics). Finally, we give necessary and sufficient conditions for $Ø_C (k)$ to be nonspecial, for any integer $k \geq 1$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 1998
 DOI:
 10.48550/arXiv.math/9805140
 arXiv:
 arXiv:math/9805140
 Bibcode:
 1998math......5140L
 Keywords:

 Mathematics  Algebraic Geometry
 EPrint:
 12 pages, to appear in Math. Scand. Mistake in earlier version of Thm 1.1 corrected and its proof is considerably simplified (removed the now redundant Sections 4 and 5 of the previous version). Added Rem. 1.2 and Prop. 1.3