Tall sections from non-minimal transformations

TL;DR

This paper reveals that certain elliptic fibrations with Mordell-Weil rank one require non-minimal transformations involving blow-ups, challenging previous assumptions of minimality in their Weierstrass models.

Contribution

It demonstrates that some Calabi-Yau elliptic fibrations with rank-one sections cannot be represented by minimal Weierstrass models and require non-minimal transformations.

Findings

Some Jacobians are not minimal Weierstrass models.

Blow-ups are necessary to embed certain fibrations into 12 models.

Recent models by Klevers and Taylor are among those requiring non-minimal transformations.

Abstract

In previous work, we have shown that elliptic fibrations with two sections, or Mordell-Weil rank one, can always be mapped birationally to a Weierstrass model of a certain form, namely, the Jacobian of a $\mathbb{P}^{112}$ model. Most constructions of elliptically fibered Calabi-Yau manifolds with two sections have been carried out assuming that the image of this birational map was a "minimal" Weierstrass model. In this paper, we show that for some elliptically fibered Calabi-Yau manifolds with Mordell-Weil rank-one, the Jacobian of the $\mathbb{P}^{112}$ model is not minimal. Said another way, starting from a Calabi-Yau Weierstrass model, the total space must be blown up (thereby destroying the "Calabi-Yau" property) in order to embed the model into $\mathbb{P}^{112}$. In particular, we show that the elliptic fibrations studied recently by Klevers and Taylor fall into this class of models.