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

Abstract

Let $f:W\to T$ 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 $Y$.

13 pages; Comments welcome; Extra assumptions were needed for the result and has been reformulated as Theorem 1 and 2; Latter sections rewritten accordingly and some arguments have been shortened