Abstract:For all integers $k>0$, we prove that the hypergeometric function \[ \widehat {I}_k(\alpha )=\sum _{j=0}^\infty \frac {\bigl ((8k+4)j\bigr )!j!}{(2j)!\bigl ((2k+1)j\bigr )!^2 \bigl ((4k+1)j\bigr )!} \ \alpha ^j \] is a period of a pencil of curves of genus $3k+1$. We prove that the function $\widehat {I}_k$ is a generating function of Gromov–Witten invariants of the family of anticanonical del Pezzo hypersurfaces $X=X_{8k+4} \subset \mathbb {P}(2,2k+1,2k+1,4k+1)$. Thus, the pencil is a Landau–Ginzburg mirror of the family. The surfaces $X$ were first constructed by Johnson and Kollár. The feature of these surfaces that makes our mirror construction especially interesting is that $|-K_X|=|\mathcal {O}_X (1)|=\varnothing$. 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.

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中文翻译：

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Johnson-Kollár del Pezzo 曲面的超椭圆积分和镜面

摘要：对于所有整数 $k>0$，我们证明超几何函数 \[ \widehat {I}_k(\alpha )=\sum _{j=0}^\infty \frac {\bigl ((8k+ 4)j\bigr )!j!}{(2j)!\bigl ((2k+1)j\bigr )!^2 \bigl ((4k+1)j\bigr )!} \ \alpha ^j \ ] 是 $3k+1$ 属的曲线的一个周期。我们证明函数 $\widehat {I}_k$ 是反规范 del Pezzo 超曲面家族的 Gromov–Witten 不变量的生成函数 $X=X_{8k+4} \subset \mathbb {P}(2,2k +1,2k+1,4k+1)$。因此，铅笔是家庭的朗道-金茨堡镜子。表面 $X$ 首先由 Johnson 和 Kollár 构建。这些表面的特征使我们的镜子构造特别有趣，因为 $|-K_X|=|\mathcal {O}_X (1)|=\​​varnothing$。这意味着无法形成 Calabi-Yau 对 $(X, D)$ 超出了 $X$，因此除了这里给出的之外，没有已知的 $X$ 镜像结构。我们还讨论了我们的构建与 Beukers、Cohen 和 Mellit 在超几何函数方面的工作之间的联系。

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