A Note on the Formal Groups of Weighted 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.