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

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.