Hyperelliptic Integrals and Mirrors of the Johnson-Kollár del Pezzo Surfaces

Abstract

For all k > 0 integer, we show explicitly that the hypergeometric functionis a period of a pencil of curves of genus 3k + 1. The function I k is the regularised I-function of the family of anticanonical del Pezzo hypersurfaces X = X 8k+4 ⊂ P(2, 2k + 1, 2k + 1, 4k + 1) and the pencil we construct is a candidate LG mirror of the elements of the family. The surfaces X were first constructed by Johnson and Kollár [17]. The main feature of these surfaces, which makes the mirror construction especially interesting, is that | − K X | = |O X (1)| = ∅; thus, there is no way to form a Calabi-Yau pair (X, D) out of X. We also discuss the connection between our constructions and the work of Beukers, Cohen and Mellit [1] on hypergeometric functions.