We present the first examples of smooth elliptic Calabi–Yau threefolds with Mordell–Weil rank 10, the highest currently known value. They are given by the Schoen threefolds introduced by Namikawa; there are six isolated fibers of Kodaira Type IV. We explicitly compute the Shioda homomorphism and the induced height pairing. Compactification of F‑theory on these threefolds gives an effective theory in six dimensions which contains ten abelian gauge group factors. We compute the massless matter spectrum. In particular, we show that the charged singlet matter need not reside at enhancement loci of Type $I_2$, as previously believed. We relate the multiplicities of the massless spectrum to genus-zero Gopakumar–Vafa invariants and other geometric quantities of the Calabi–Yau. We show that the gravitational and abelian anomaly cancellation conditions are satisfied. We prove a Geometric Anomaly Cancellation equation and we deduce birational equivalence for the quantities in the spectrum. We explicitly describe a Weierstrass model over $\mathbb{P}^2$ of the Calabi–Yau threefolds as a log canonical model and compare it to a construction by Elkies and classical results of Burkhardt.

中文翻译：

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具有高 Mordell-Weil 秩的椭圆三重

我们展示了第一个平滑椭圆 Calabi-Yau 三倍的例子，Mordell-Weil 等级为 10，这是目前已知的最高值。它们是由波川介绍的 Schoen 三重给予的；Kodaira IV 型有 6 根孤立的纤维。我们明确计算 Shioda 同态和诱导高度配对。F 理论在这三重上的紧化给出了一个有效的六维理论，其中包含十个阿贝尔规范群因子。我们计算无质量物质谱。特别是，我们表明带电单线态物质不必像以前认为的那样位于类型 $I_2$ 的增强基因座上。我们将无质量谱的多重性与零属 Gopakumar-Vafa 不变量和 Calabi-Yau 的其他几何量联系起来。我们证明了引力和阿贝尔异常消除条件是满足的。我们证明了几何异常消除方程，并推导出谱中的量的双有理等价。我们明确地描述了一个在 Calabi-Yau 三重的 $\mathbb{P}^2$ 上的 Weierstrass 模型作为对数规范模型，并将其与 Elkies 的构造和 Burkhardt 的经典结果进行比较。