Quotients of Hypersurfaces in Weighted Projective Space

TL;DR

This paper generalizes a construction relating quotients of Calabi-Yau varieties in weighted projective spaces to Fermat varieties, demonstrating birational equivalences among certain Calabi-Yau families.

Contribution

It introduces a new framework for constructing quotients of hypersurfaces in weighted projective space using invertible matrices, extending previous methods.

Findings

$ar{M}_A$ is a quotient of a Fermat variety by a finite group

$X_A$ is a quotient of a Fermat variety

Certain Calabi-Yau families are shown to be birational

Abstract

In [1] some quotients of one-parameter families of Calabi-Yau varieties are related to the family of Mirror Quintics by using a construction due to Shioda. In this paper, we generalize this construction to a wider class of varieties. More specifically, let $A$ be an invertible matrix with non-negative integer entries. We introduce varieties $X_A$ and $\overline{M}_A$ in weighted projective space and in ${\mathbb P}^n$, respectively. The variety $\overline{M}_A$ turns out to be a quotient of a Fermat variety by a finite group. As a by-product, $X_A$ is a quotient of a Fermat variety and $\overline{M}_A$ is a quotient of $X_A$ by a finite group. We apply this construction to some families of Calabi-Yau manifolds in order to show their birationality.