Non-minimal Elliptic Threefolds at Infinite Distance I: Log Calabi-Yau Resolutions

TL;DR

The authors develop a geometric program to study infinite-distance limits in the complex structure moduli space of elliptic Calabi–Yau threefolds relevant to 6D F-theory, focusing on degenerations with non-minimal Kodaira fibers. They resolve these non-crepant singularities by a systematic sequence of base blow-ups that produce a reducible, gluable union of log Calabi–Yau components, each hosting specialized Weierstrass data. They classify codimension-one degenerations into five classes, prove that Class 5 can be removed via semi-stable reduction (potentially after base change), and show that single infinite-distance limits give open-chain resolution geometries with Hirzebruch-component bases. The paper then specializes to Hirzebruch bases to illustrate horizontal, vertical, and mixed degenerations, detailing the line-bundle data, discriminant structure, and restrictions on genus-zero curves that support non-minimal fibers. A core contribution is a robust algorithm to extract the codimension-one gauge algebra from the multi-component central fiber, accounting for discriminant gluing and monodromy; the work lays the groundwork for a companion paper that connects these geometric structures to explicit physical interpretations, including decompactification and heterotic duality. The results provide a precise bridge between complex-structure degeneration geometry and F-theory gauge data in infinite-distance limits, enabling systematic analysis of asymptotic physics in six dimensions.

Abstract

We study infinite-distance limits in the complex structure moduli space of elliptic Calabi-Yau threefolds. In F-theory compactifications to six dimensions, such limits include infinite-distance trajectories in the non-perturbative open string moduli space. The limits are described as degenerations of elliptic threefolds whose central elements exhibit non-minimal elliptic fibers, in the Kodaira sense, over curves on the base. We show how these non-crepant singularities can be removed by a systematic sequence of blow-ups of the base, leading to a union of log Calabi-Yau spaces glued together along their boundaries. We identify criteria for the blow-ups to give rise to open chains or more complicated trees of components and analyse the blow-up geometry. While our results are general and applicable to all non-minimal degenerations of Calabi-Yau threefolds in codimension one, we exemplify them in particular for elliptic threefolds over Hirzebruch surface base spaces. We also explain how to extract the gauge algebra for F-theory probing such reducible asymptotic geometries. This analysis is the basis for a detailed F-theory interpretation of the associated infinite-distance limits that will be provided in a companion paper.

Figures (18)

• Figure 1: A representation of a semi-stable degeneration of elliptically fibered threefolds. The blue disk represents $D$, over which we have two generic fibers $Y_{u_{1}}$ and $Y_{u_{2}}$, that degenerate to the multi-component central fiber $Y_{0}$.
• Figure 2: The open-chain of $B^{p}$ components arising for the central fiber $B_{0}$ of the base family variety $\mathcal{B}$ of the (open-chain) resolution of a single infinite-distance limit degeneration $\hat{\rho}: \hat{\mathcal{Y}} \rightarrow D$. The strict transform $B^{0}$ of the original base $\hat{B}$ preserves its original geometry $B^{0} \cong \hat{B}_{0}$, while the rest of the components $B^{p}$ for $1 \leq p \leq P$ are Hirzebruch surfaces, as proved in \ref{['prop:component-geometry-single']}.
• Figure 3: Central fiber $B_{0}$ of the base $\mathcal{B}$ of the tree resolution of a general degeneration $\hat{\rho}: \hat{\mathcal{Y}} \rightarrow D$ not falling under the single infinite-distance limit category. Note that a resolution with the configuration of components depicted here will always present obscured infinite-distance limits, see the discussions in \ref{['sec:obscured-infinite-distance-limits', 'sec:single-infinite-distance-limits-and-open-chain-resolutions']}.
• Figure 4: Toric fans associated to the family base of a horizontal model.
• Figure 5: Restrictions $\Delta'_{0}$ and $\Delta'_{1}$ of the (modified) discriminant for \ref{['example:illustrative-example']}, with the residual discriminant omitted for clarity. The printed vanishing orders correspond to the component vanishing orders in each component.

Theorems & Definitions (85)

• Example 2.1
• Definition 2.2: Family vanishing orders
• Definition 2.3: Component vanishing orders
• Definition 2.4: Interface vanishing orders
• Definition 2.5: Degenerations of Class 1--5
• Theorem 2.6: Semi-stable Reduction Theorem Mumford1973
• Proposition 2.6
• Corollary 2.7
• Definition 2.7: Single infinite-distance limits
• Definition 2.8: Open-chain resolution