A result on the generic Picard number of surfaces in fake weighted projective 3-spaces

TL;DR

This paper establishes a criterion for certain algebraic surfaces in fake weighted projective 3-spaces to have Picard number greater than one, using degenerations and cohomology techniques.

Contribution

It introduces a new criterion for the Picard number of surfaces in fake weighted projective spaces, focusing on degenerations and cohomological methods.

Findings

Identifies conditions under which surfaces have Picard number > 1

Uses degenerations along edges to analyze geometric genus

Constructs rational Picard classes not proportional to the canonical divisor

Abstract

We give a criterion for certain generic nondegenerate surfaces in a fake weighted projective $3$-space to have Picard number $>1$. These algebraic surfaces are of general type. We do this by considering degenerations (along an edge), keeping track of the geometric genus, and using vanishing cohomology classes to construct a rational Picard class on the surface not proportional to the canonical divisor.