Noether–Lefschetz theory with base locus

Abstract

Let $$Z$$ be a closed subscheme of a smooth complex projective variety $$Y\subseteq \mathbb {P}^N$$ , with $$\dim \,Y=2r+1\ge 3$$ . We describe the intermediate Néron–Severi group (i.e. the image of the cycle map $$A_r(X)\rightarrow H_{2r}(X;\mathbb {Z})$$ ) of a general smooth hypersurface $$X\subset Y$$ of sufficiently large degree containing $$Z$$ .