On the birationality of the adjunction mapping of projective varieties

Let $X$ be a smooth projective $n$-fold such that $q(X)=0$ and $L$ a globally generated, big line bundle on $X$ such that $h^0(K_X+(n-2)L) >0$. We give necessary and sufficient conditions for the adjoint systems $|K_X+kL|$ to be birational for $k \geq n-1$. In particular, for Calabi-Yau $n$-folds we generalize and prove parts of a conjecture of Gallego and Purnaprajna.Comment: 7 pages, accepted for publication in Journal of the Ramanujan Mathematical Societ

Remarks on families of singular curves with hyperelliptic normalizations

We give restrictions on the existence of families of curves on smooth projective surfaces $S$ of nonnegative Kodaira dimension all having constant geometric genus $g \geq 2$ and hyperelliptic normalizations. In particular, we prove a Reider-like result whose proof is ``vector bundle-free'' and relies on deformation theory and bending-and-breaking of rational curves in \Sym^2(S). We also give examples of families of such curves.Comment: 18 page

Smooth curves on projective K3 surfaces

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 \O_C (k) to be non-special, for any integer $k \geq 1$.Comment: 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.