A New Model for Elliptic Fibrations with a Rank One Mordell-Weil Group: I. Singular Fibers and Semi-Stable Degenerations

TL;DR

This paper introduces a new elliptic fibration model with a rank one Mordell-Weil group, enabling explicit computations of singular fibers, degenerations, and topological invariants relevant for F-theory and string theory.

Contribution

It presents the Q$_7( ext{L}, ext{S})$ model, generalizing previous models, and analyzes its singular fibers, degenerations, and topological properties in the context of F-theory.

Findings

Determined the spectrum of singular fibers for the model.

Computed a generating function for the Euler characteristic.

Established a topological relation ensuring D3 charge consistency.

Abstract

We introduce a new model for elliptic fibrations endowed with a Mordell-Weil group of rank one. We call it a Q$_7(\mathscr{L},\mathscr{S})$ model. It naturally generalizes several previous models of elliptic fibrations popular in the F-theory literature. The model is also explicitly smooth, thus relevant physical quantities can be computed in terms of topological invariants in straight manner. Since the general fiber is defined by a cubic curve, basic arithmetic operations on the curve can be done using the chord-tangent group law. We will use this model to determine the spectrum of singular fibers of an elliptic fibration of rank one and compute a generating function for its Euler characteristic. With a view toward string theory, we determine a semi-stable degeneration which is understood as a weak coupling limit in F-theory. We show that it satisfies a non-trivial topological relation at the level of homological Chern classes. This relation ensures that the D3 charge in F-theory is the same as the one in the weak coupling limit.