Noether-Lefschetz Theory with base locus

TL;DR

This paper investigates the intermediate Néron-Severi group of general smooth hypersurfaces in complex projective varieties containing a fixed subscheme, extending Noether-Lefschetz theory to include base loci.

Contribution

It provides a description of the intermediate Néron-Severi group for hypersurfaces with a fixed base locus in higher-dimensional varieties, generalizing classical results.

Findings

Explicit characterization of the intermediate Néron-Severi group

Extension of Noether-Lefschetz theory to varieties with base locus

Results for hypersurfaces of large degree containing a fixed subscheme

Abstract

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