Smooth curves on projective K3 surfaces
Abstract
Description
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 non-special, for any integer $k \geq 1$.
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
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