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Does anyone know any example of two elliptically fibered toric variety (3-fold) that are birational to each other?

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Over $\mathbb{C}$ all toric varieties are rational, hence any two (of the same dimension) are birational. Many examples of toric 3-folds with elliptic fibrations are given in the following paper https://arxiv.org/pdf/1110.4883.pdf.

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