# Hyperelliptic Integrals and Mirrors of the Johnson-Koll\'ar del Pezzo Surfaces

## TL;DR

This paper constructs explicit Landau-Ginzburg mirror models for a family of del Pezzo surfaces with empty anticanonical systems, using hypergeometric functions as periods of specific curve pencils.

## Contribution

It introduces a novel mirror construction for del Pezzo surfaces with empty anticanonical systems via hypergeometric periods, expanding mirror symmetry methods.

## Key findings

- The I-function is a period of an explicit pencil of curves.
- The constructed pencil serves as a candidate LG mirror.
- Connections to hypergeometric function theory are established.

## Abstract

For all k>0 integer, we consider the regularised I-function of the family of del Pezzo surfaces of degree 8k+4 in P(2,2k+1,2k+1, 4k+1), first constructed by Johnson and Koll\'ar. We show that this function, which is of hypergeometric type, is a period of an explicit pencil of curves. Thus the pencil is a candidate LG mirror of the family of del Pezzo surfaces. The main feature of these surfaces, which makes the mirror construction especially interesting, is that the anticanonical system is empty: because of this, our mirrors are not covered by any other construction known to us. We discuss connections to the work of Beukers, Cohen and Mellit on hypergeometric functions.