Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 097, 12 pages      arXiv:1805.04233      https://doi.org/10.3842/SIGMA.2018.097
Contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui

A Note on the Formal Groups of Weighted Delsarte Threefolds

Yasuhiro Goto
Department of Mathematics, Hokkaido University of Education, 1-2 Hachiman-cho, Hakodate 040-8567 Japan

Received May 11, 2018, in final form August 30, 2018; Published online September 12, 2018

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.

Key words: Artin-Mazur formal groups; Calabi-Yau threefolds; weighted Delsarte varieties.

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