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

David Wen

Published 2021-04-23Version 1

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 $(f,g,4f^2 + 27g^3)$ vanishes of order $(4,6,12)$ over an isolated point over $T$. Such a fiber can be resolved to a terminal model, $Y$, containing 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 bound of the free rank of Mordell-Weil group of $W \rightarrow T$ below by the free rank of the Mordell-Weil group of $S$.