A Note on the Formal Groups of Weighted Delsarte Threefolds

TL;DR

This paper investigates the formal groups of weighted Delsarte Calabi-Yau threefolds in positive characteristic, providing an algorithm to compute their height and analyzing several examples to understand their distribution.

Contribution

It introduces a new algorithm for determining the height of formal groups of weighted Delsarte threefolds and applies it to various examples, expanding understanding of their properties.

Findings

Algorithm successfully computes formal group heights.

Distribution of heights varies among examples.

Provides new data on formal groups of Delsarte threefolds.

Abstract

One-dimensional formal groups over an algebraically closed field of positive characteristic are classified by their height. In the case of $K3$ surfaces, the height of their formal groups takes integer values between $1$ and $10$, or $\infty$. For Calabi-Yau threefolds, the height is bounded by $h^{1,2}+1$ if it is finite, where $h^{1,2}$ is a Hodge number. At present, there are only a limited number of concrete examples for explicit values or the distribution of the height. In this paper, we consider Calabi-Yau threefolds arising from weighted Delsarte threefolds in positive characteristic. We describe an algorithm for computing the height of their formal groups and carry out calculations with various Calabi-Yau threefolds of Delsarte type.