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

TL;DR

This paper investigates elliptic threefolds with specific singular fibers, exploring how their geometric resolutions relate to the Mordell-Weil group and flopping curves, revealing new constraints and connections.

Contribution

It provides a detailed analysis of the relationship between singular fiber resolutions, flops, and the Mordell-Weil group in elliptic threefolds with (4,6,12)-singular fibers.

Findings

Explicit resolution of singular fibers with smaller vanishing order.

Identification of constraints between flops and Mordell-Weil group properties.

Connection between rational elliptic surfaces and flopping curves.

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 $Y$.