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Weierstrass models of elliptic toric K3 hypersurfaces and symplectic cuts

Antonella Grassi, Vittorio Perduca

TL;DR

The paper develops a toric framework to study elliptically fibered $K3$ surfaces inside toric Fano threefolds, introducing toric Weierstrass models and semistable degenerations compatible with F-theory/Heterotic duality. By recasting Candelas' combinatorial conditions into semistable polytopes and infinity sections, it proves that under these conditions the ambient toric threefold and the $K3$ hypersurface admit a semistable degeneration via a symplectic cut to two rational elliptic surfaces glued along the elliptic fiber, while preserving the fibration. It provides explicit criteria for the existence of a section at infinity and constructs Candelas–Font Weierstrass models within a toric ambient, connecting the geometry to $E_8 imes E_8$ dual data. Overall, the work offers a rigorous, combinatorial route to realize F-theory/Heterotic duality in the toric setting and lays groundwork for higher-dimensional extensions via toric degenerations.

Abstract

We study elliptically fibered K3 surfaces, with sections, in toric Fano threefolds which satisfy certain combinatorial properties relevant to F-theory/Heterotic duality. We show that some of these conditions are equivalent to the existence of an appropriate notion of a Weierstrass model adapted to the toric context. Moreover, we show that if in addition other conditions are satisfied, there exists a toric semistable degeneration of the elliptic K3 surface which is compatible with the elliptic fibration and F-theory/Heterotic duality.

Weierstrass models of elliptic toric K3 hypersurfaces and symplectic cuts

TL;DR

The paper develops a toric framework to study elliptically fibered surfaces inside toric Fano threefolds, introducing toric Weierstrass models and semistable degenerations compatible with F-theory/Heterotic duality. By recasting Candelas' combinatorial conditions into semistable polytopes and infinity sections, it proves that under these conditions the ambient toric threefold and the hypersurface admit a semistable degeneration via a symplectic cut to two rational elliptic surfaces glued along the elliptic fiber, while preserving the fibration. It provides explicit criteria for the existence of a section at infinity and constructs Candelas–Font Weierstrass models within a toric ambient, connecting the geometry to dual data. Overall, the work offers a rigorous, combinatorial route to realize F-theory/Heterotic duality in the toric setting and lays groundwork for higher-dimensional extensions via toric degenerations.

Abstract

We study elliptically fibered K3 surfaces, with sections, in toric Fano threefolds which satisfy certain combinatorial properties relevant to F-theory/Heterotic duality. We show that some of these conditions are equivalent to the existence of an appropriate notion of a Weierstrass model adapted to the toric context. Moreover, we show that if in addition other conditions are satisfied, there exists a toric semistable degeneration of the elliptic K3 surface which is compatible with the elliptic fibration and F-theory/Heterotic duality.

Paper Structure

This paper contains 15 sections, 15 theorems, 27 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and notations
  3. Toric, Fano varieties
  4. Calabi-Yau varieties and reflexive polytopes
  5. Intersection on toric $K3$ hypersurfaces
  6. Elliptic Fibrations
  7. Candelas' Examples
  8. Semistable polytopes, Sections at infinity, Weierstass models (Candelas' conditions recasted).
  9. Semistable polytopes
  10. Infinity sections, toric flexes
  11. Candelas-Fonts Weierstrass models
  12. Toric and non toric sections
  13. Symplectic cut, degenerations, physics duality
  14. Equations of elliptic curves in toric Del Pezzo surfaces
  15. Ellitpic fibration in homogeneous coordinates

Key Result

Theorem 2.3

If $\Delta\subset M_{\mathbb{R}}\simeq\mathbb{R}^n$ is a reflexive polytope of dimension $n$, then the general member $\bar{V}\in|-K_{\mathbb{P}_{\Delta}}|$ is a Calabi-Yau variety of dimension $n-1$. If $\Sigma$ is a projective subdivision of the normal fan of $\Delta$, then

Figures (3)

  • Figure 1: Reflexive polytopes in the plane
  • Figure 2: The polytope $\nabla \subset N_{\mathbb{R}}$
  • Figure 3: The polytope $\Delta\subset M_{\mathbb{R}}$ dual to $\nabla \subset N_{\mathbb{R}}$

Theorems & Definitions (45)

  • Theorem 2.3: Ch. 4 CoKa99
  • Remark 2.4
  • Theorem 2.5: PeSk97
  • Example 3.1: Polytope 3737
  • Example 3.2: Polytope 4318 fibered by 9
  • Example 3.3: Polytope 113: "Diamond" fibered by 15
  • Example 3.4: Polyotope 4: "Diamond" fibered by 1
  • Definition 4.1
  • Theorem 4.3
  • proof
  • ...and 35 more