Grothendieck-Lefschetz Theorem with Base Locus

TL;DR

This paper computes the divisor class group of general hypersurfaces in complex projective varieties with fixed base loci, revealing that certain local complete intersection domains are completions of unique factorization domains.

Contribution

It provides a new method to determine the divisor class group of hypersurfaces with base loci and links local complete intersection domains to UFD completions.

Findings

Divisor class group of general hypersurfaces computed

Normal local complete intersection domains are UFD completions

Results apply to complex projective varieties of dimension at least four

Abstract

We compute the divisor class group of the general hypersurface Y of a complex projective normal variety X of dimension at least four containing a fixed base locus Z. We deduce that completions of normal local complete intersection domains of finite type over the complex numbers of dimension $\ge 4$ are completions of UFDs of finite type over the complex numbers.