Tall sections from non-minimal transformations

TL;DR

This work analyzes elliptic fibrations with Mordell-Weil rank-one in F-theory and shows Klevers–Taylor models arise from non-minimal birational maps to Weierstrass MP-form models. The KT Jacobian is not minimal, and the CY fibration can be obtained from a non-CY parent via a blowdown, producing taller generating sections and enabling matter with charges up to $3$. The authors provide explicit expressions for the KT data $a,b,c_i$ in terms of KT's $s_i$, establishing a two-parameter family of solutions and proving a precise birational relation to the MP form. Overall, the results broaden the landscape of MW rank-one Calabi–Yau fibrations and illuminate how non-minimal embeddings influence abelian charges and anomaly coefficients in F-theory model-building.

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.