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@article{Corti:2021:tran/8465,
author = {Corti, A and Gugiatti, G},
doi = {tran/8465},
journal = {Transactions of the American Mathematical Society},
pages = {8603--8637},
title = {Hyperelliptic integrals and mirrors of the Johnson-Kollár del Pezzo surfaces},
url = {http://dx.doi.org/10.1090/tran/8465},
volume = {374},
year = {2021}
}

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TY  - JOUR
AB  - For all integers k > 0, we prove that the hypergeometric functionIbk(α) = X∞ j=0  (8k + 4)j [divided by] !j! (2j)! (2k + 1)j!2 (4k + 1)j! αjis a period of a pencil of curves of genus 3k+1. We prove that the function Ibk is a generating function of Gromov–Witten invariants of the family of anticanonical del Pezzo hypersurfaces X = X8k+4 ⊂ P(2, 2k + 1, 2k + 1, 4k + 1). Thus, the pencil is a Landau–Ginzburg mirror of the family. The surfaces X were first constructed by Johnson and Koll´ar. The feature of these surfaces that makes our mirror construction especially interesting is that | − KX| = |OX(1)| = ∅. This means that there is no way to form a Calabi–Yau pair (X, D) out of X and hence there is no known mirror construction for X other than the one given here. We also discuss the connection between our construction and work of Beukers, Cohen and Mellit on hypergeometric functions.
AU  - Corti,A
AU  - Gugiatti,G
DO  - tran/8465
EP  - 8637
PY  - 2021///
SN  - 0002-9947
SP  - 8603
TI  - Hyperelliptic integrals and mirrors of the Johnson-Kollár del Pezzo surfaces
T2  - Transactions of the American Mathematical Society
UR  - http://dx.doi.org/10.1090/tran/8465
UR  - https://www.ams.org/journals/tran/earlyview/#tran8465
UR  - http://hdl.handle.net/10044/1/90215
VL  - 374
ER  -

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