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

Alessio Corti; Giulia Gugiatti

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arXiv · 1901.09026

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

Alessio Corti·Giulia Gugiatti

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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.

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Alessio Corti, Giulia Gugiatti. 2019-01-25. Hyperelliptic Integrals and Mirrors of the Johnson-Koll\'ar del Pezzo Surfaces. https://arxiv.org/abs/1901.09026

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