Points in projective spaces and applications

TL;DR

This paper establishes conditions under which certain nodal hypersurfaces and double covers in projective spaces are factorial, based on bounds on their singular points, contributing to algebraic geometry's understanding of factoriality.

Contribution

It proves factoriality of specific nodal hypersurfaces and double covers in projective spaces with bounds on singularities, extending existing factoriality criteria.

Findings

Factoriality of a nodal hypersurface in P^4 with ≤ 2(d-1)^2/3 singular points.

Factoriality of a double cover of P^3 branched over a nodal surface with less than (2r-1)r singular points.

Abstract

We prove the factoriality of a nodal hypersurface in $\mathbb{P}^{4}$ of degree $d$ that has at most $2(d-1)^{2}/3$ singular points, and factoriality of a double cover of $\mathbb{P}^{3}$ branched over a nodal surface of degree $2r$ having less than $(2r-1)r$ singular points.