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

TL;DR

The paper introduces the Q$_7(\mathscr{L},\mathscr{S})$ model for elliptic fibrations with a rank-one Mordell–Weil group, providing a smooth cubic-fiber framework that unifies and extends existing models while enabling direct computation of topological invariants via the chord–tangent law. It derives the full spectrum of singular fibers (including a non-Kodaira IV$^{(2)}$ fiber) and proves a generalized Sethi–Vafa–Witten Euler-characteristic formula through a pushforward of the total Chern class. A central result is a semi-stable degeneration realizing a weak coupling limit in F-theory, accompanied by a topological tadpole relation at the level of total Chern classes, ensuring D3-charge matching with the orientifold limit and providing a clean flux-matching framework. The work advances the geometric and topological control over F-theory compactifications with $U(1)$ gauge symmetry, enabling precise comparisons to type IIB weak coupling descriptions and guiding future studies of brane spectra, fluxes, and dualities in fourfold geometries.

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.

Figures (3)

• Figure 1: Newton polygon of a Q$_7$ reflexive polytope. A Q$_7(\mathscr{L},\mathscr{S})$ model is described by a section of a line bundle $O(3)\otimes \pi^{\ast} \mathscr{L}^2\otimes \pi^{\ast}\mathscr{S}$ in the projective bundle $\mathbb{P}[\mathscr{L}^2\oplus \mathscr{S}\oplus\mathscr{O}_B].$ Its equation is automatically of type Q$_7$.
• Figure 2: Singular fibers of plane cubic curves. There are a total of 8 possible singular fibers including the 6 Kodaira fibers with at most 3 components ($I_1, II, I_2, III, I_3, IV$) and the two non-Kodaira fibers $IV^{(2)}$ and $IV^{(3)}$. All the fibers at the left of a given dotted vertical line are those of a smooth elliptic fibration of the type ($E_8, E_7, E_6, Q_7)$ specified at the bottom left of the dotted line.
• Figure 3: Quartic Q$_7$: a reflexive quadrilateral with seven lattice points on its boundary. This is the Newton's polygon for the quartic in equation \ref{['Equation.quartic']}.

Theorems & Definitions (16)

• Remark 1.1
• Theorem 1.2: Topological tadpole matching for Q$_7(\mathscr{L},\mathscr{S})$ elliptic fibrations
• Theorem 2.1: Calabi-Yau condition
• Remark 2.2
• Theorem 2.3: Mordell-Weil group
• Theorem 2.4
• Lemma 2.5
• Remark 2.6
• Theorem 2.7: Euler characteristic of Q$_7(\mathscr{L},\mathscr{S})$
• Lemma 2.8