Communications in Number Theory and Physics

Volume 13 (2019)

Number 1

Picard–Fuchs operators for octic arrangements, I: The case of orphans

Pages: 1 – 52



Slawomir Cynk (Institute of Mathematics, Jagiellonian University, Kraków, Poland; and Institute of Mathematics of the Polish Academy of Sciences, Warszawa, Poland)

Duco van Straten (Algebraische Geometrie, Johannes Gutenberg-Universität, Mainz, Germany)


We report on 25 families of projective Calabi–Yau threefolds that do not have a point of maximal unipotent monodromy in their moduli space. The construction is based on an analysis of certain pencils of octic arrangements that were found by C. Meyer. There are seven cases where the Picard–Fuchs operator is of order two and 18 cases where it is of order four. The birational nature of the Picard–Fuchs operator can be used effectively to distinguish between families whose members have the same Hodge numbers.

This research was supported in part by PL-Grid Infrastructure.

The first author was partly supported by the Schwerpunkt Polen (Mainz) and NCN grant no. N N201 608040.

The second author was partly supported by DFG Sonderforschungsbereich/Transregio 45.

Received 29 September 2017

Accepted 9 August 2018

Published 29 April 2019