Noether-Lefschetz Theory and N\'eron-Severi Group

TL;DR

This paper investigates the Néron-Severi group of general hypersurfaces within smooth complex projective varieties, providing descriptions for cases with large degrees and specific subschemes.

Contribution

It offers a new description of the Néron-Severi group for hypersurfaces containing a given subscheme in high-degree cases within complete intersections.

Findings

Explicit description of NS_r(X) for large degree hypersurfaces

Conditions under which the Néron-Severi group can be computed

Extension of Noether-Lefschetz theory to higher codimension

Abstract

Let $Z$ be a closed subscheme of a smooth complex projective complete intersection variety $Y\subseteq \Ps^N$, with $dim Y=2r+1\geq 3$. We describe the N\'eron-Severi group $NS_r(X)$ of a general smooth hypersurface $X\subset Y$ of sufficiently large degree containing $Z$.