Bounds on Fake Weighted Projective Space

TL;DR

This paper investigates the properties of fake weighted projective spaces, focusing on how their singularities relate to weights and establishing bounds on possible configurations for fixed dimensions.

Contribution

It provides new bounds on the weights and the number of fake weighted projective spaces with certain singularity types, enhancing understanding of their structure.

Findings

Singularities of fake weighted projective spaces are influenced by those of weighted projective spaces.

Bounds are established on the number of such spaces for a given dimension.

Upper bounds on weight ratios are derived for terminal and canonical singularities.

Abstract

A fake weighted projective space X is a Q-factorial toric variety with Picard number one. As with weighted projective space, X comes equipped with a set of weights (\lambda_0,...,\lambda_n). We see how the singularities of P(\lambda_0,...,\lambda_n) influence the singularities of X, and how the weights bound the number of possible fake weighted projective spaces for a fixed dimension. Finally, we present an upper bound on the ratios \lambda_j/\sum\lambda_i if we wish X to have only terminal (or canonical) singularities.