Tall sections from 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.

16 pages

References in corpus (19)

• Evidence for F-Theory
• Compactifications of F-Theory on Calabi--Yau Threefolds -- II
• Compactifications of F-Theory on Calabi--Yau Threefolds -- I
• F-Theory and the Mordell-Weil Group of Elliptically-Fibered Calabi-Yau Threefolds
• F-Theory on all Toric Hypersurface Fibrations and its Higgs Branches
• Point-like Instantons on K3 Orbifolds
• Anomaly Equations and Intersection Theory
• F-Theory Compactifications with Multiple U(1)-Factors: Constructing Elliptic Fibrations with Rational Sections
• Geometric Engineering in Toric F-Theory and GUTs with U(1) Gauge Factors
• Anomaly Cancellation And Abelian Gauge Symmetries In F-theory
• U(1) symmetries in F-theory GUTs with multiple sections
• SU(5) Tops with Multiple U(1)s in F-theory
• General U(1)xU(1) F-theory Compactifications and Beyond: Geometry of unHiggsings and novel Matter Structure
• F-theory and All Things Rational: Surveying U(1) Symmetries with Rational Sections
• A New Model for Elliptic Fibrations with a Rank One Mordell-Weil Group: I. Singular Fibers and Semi-Stable Degenerations
• Tate Trees for Elliptic Fibrations with Rank one Mordell-Weil group
• F-GUTs with Mordell-Weil U(1)'s
• Heterotic T-fects, 6D SCFTs, and F-Theory
• Three-Index Symmetric Matter Representations of SU(2) in F-Theory from Non-Tate Form Weierstrass Models

Cited by in corpus (22)

• Tensionless Strings and the Weak Gravity Conjecture
• 6d SCFTs and U(1) Flavour Symmetries
• Non-Higgsable abelian gauge symmetry and F-theory on fiber products of rational elliptic surfaces
• Discrete Gauge Groups in F-theory Models on Genus-One Fibered Calabi-Yau 4-folds without Section
• Enhancements in F-theory models on moduli spaces of K3 surfaces with $ADE$ rank 17
• Exotic matter on singular divisors in F-theory
• Discrete gauge groups in certain F-theory models in six dimensions
• Abelian F-theory Models with Charge-3 and Charge-4 Matter
• Nongeometric heterotic strings and dual F-theory with enhanced gauge groups
• Tuned and Non-Higgsable U(1)s in F-theory
• High electric charges in M-theory from quiver varieties
• When Rational Sections Become Cyclic: Gauge Enhancement in F-theory via Mordell--Weil Torsion
• Unbroken $E_7\times E_7$ nongeometric heterotic strings, stable degenerations and enhanced gauge groups in F-theory duals
• F-theory models with 3 to 8 U(1) factors on K3 surfaces
• F-theory models with $U(1)\times {\mathbb{Z}}_2,{\mathbb{Z}}_4$ and transitions in discrete gauge groups
• Orders of Vanishing and U(1) Charges in F-theory
• $\frac{1}{2}$ Calabi-Yau 3-folds, Calabi-Yau 3-folds as double covers, and F-theory with U(1)s
• Types of gauge groups in six-dimensional F-theory on double covers of rational elliptic 3-folds
• A note on transition in discrete gauge groups in F-theory
• Extremal 1/2 Calabi-Yau 3-folds and six-dimensional F-theory applications
• $\frac{1}{2}$Calabi-Yau 4-folds and four-dimensional F-theory on Calabi-Yau 4-folds with U(1) factors
• Singularities of 1/2 Calabi-Yau 4-folds and classification scheme for gauge groups in four-dimensional F-theory