Flops and Mordell-Weil group of Elliptic Threefolds with (4,6,12)-singular fibers

David Wen

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arXiv · 2104.11443

Flops and Mordell-Weil group of Elliptic Threefolds with (4,6,12)-singular fibers

David Wen

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Abstract

Let $f: W \rightarrow T$ be an elliptic threefold that is a Weierstrass model, which is locally defined by $y^2 = x^3 + fx + g$ over $T$, with a singular fiber such that $(f,g,4f^3 + 27g^2)$ vanishes of order $(4,6,12)$ over an isolated point over $T$. Such a fiber can be explicitly resolved to fibers with smaller vanishing order and the resulting model, $Y$, contains a rational elliptic surface, $S$, where some sections of $S$ are flopping curves on $Y$. As a consequence of this arithmetic and geometric connection, we are able to describe some constraints between flops of $Y$ and the properties of the Mordell-Weil group of $S$ and $Y$.

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David Wen. 2021-04-23. Flops and Mordell-Weil group of Elliptic Threefolds with (4,6,12)-singular fibers. https://arxiv.org/abs/2104.11443

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