1998/05/29 by Andreas Leopold Knutsen, Knutsen, Andreas Leopold
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG
paper · pdf · doi:10.48550/arxiv.math/9805140
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
arxiv created 2001/06/07 · arxiv updated 2009/11/30
In this paper we give for all n ≥ 2, d>0, g ≥ 0 necessary and sufficient conditions for the existence of a pair (X,C), where X is a K3 surface of degree 2n in \matbfPn+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 ≥ 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 ØC (k) to be non-special, for any integer k ≥ 1.