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Calabi-Yau complete intersections associated to good pairs of generalized nef partitions

Michela Artebani, Paola Comparin, Robin Guilbot

TL;DR

This work extends nef-partition theory to generalized nef partitions in ${\mathbb Q}$-Fano toric varieties and introduces good pairs, along with a duality that unifies and extends Batyrev–Borisov mirror symmetry and Berglund–Hübsch–Krawitz-type dualities. It builds a cohesive combinatorial and toric-geometric framework linking polytopes to nef divisors, enabling construction of Calabi–Yau complete intersections via quasismooth data and a robust duality that yields mirror families, including Delsarte-type and fake weighted projective ambient spaces. The paper also provides practical criteria for quasismoothness through the Cayley trick, gives explicit examples (notably codimension-two K3 surfaces), and supplies MAGMA tools to compute and verify generalized nef partitions and their duals. Overall, it broadens the landscape of toric Calabi–Yau constructions and mirrors beyond maximal Newton polytopes, offering new pathways for exploring mirror symmetry in toric and orbifold settings.

Abstract

We introduce the notion of good pair of generalized nef partitions to describe Calabi-Yau complete intersections in Q-Fano toric varieties whose equations do not necessarily have maximal Newton polytopes. Moreover, we define a natural duality between them which generalizes Batyrev-Borisov mirror duality and allows to define a generalization of Berglund-Hübsch-Krawitz duality to quasismooth complete intersections.

Calabi-Yau complete intersections associated to good pairs of generalized nef partitions

TL;DR

This work extends nef-partition theory to generalized nef partitions in -Fano toric varieties and introduces good pairs, along with a duality that unifies and extends Batyrev–Borisov mirror symmetry and Berglund–Hübsch–Krawitz-type dualities. It builds a cohesive combinatorial and toric-geometric framework linking polytopes to nef divisors, enabling construction of Calabi–Yau complete intersections via quasismooth data and a robust duality that yields mirror families, including Delsarte-type and fake weighted projective ambient spaces. The paper also provides practical criteria for quasismoothness through the Cayley trick, gives explicit examples (notably codimension-two K3 surfaces), and supplies MAGMA tools to compute and verify generalized nef partitions and their duals. Overall, it broadens the landscape of toric Calabi–Yau constructions and mirrors beyond maximal Newton polytopes, offering new pathways for exploring mirror symmetry in toric and orbifold settings.

Abstract

We introduce the notion of good pair of generalized nef partitions to describe Calabi-Yau complete intersections in Q-Fano toric varieties whose equations do not necessarily have maximal Newton polytopes. Moreover, we define a natural duality between them which generalizes Batyrev-Borisov mirror duality and allows to define a generalization of Berglund-Hübsch-Krawitz duality to quasismooth complete intersections.
Paper Structure (17 sections, 24 theorems, 96 equations, 4 figures, 3 tables)

This paper contains 17 sections, 24 theorems, 96 equations, 4 figures, 3 tables.

Table of Contents

  1. Toric background
  2. Toric setting
  3. Laurent systems and their homogenization
  4. Regularity conditions
  5. Quasismoothness
  6. Generalized nef partitions
  7. Definition and first properties
  8. Duality
  9. Irreducibility
  10. Toric geometry of generalized nef partitions
  11. Good pairs and Calabi-Yau complete intersections
  12. Good pairs of generalized nef partitions
  13. Calabi-Yau varieties associated to good pairs
  14. Duality
  15. From equations to good pairs of generalized nef partitions
  16. ...and 2 more sections

Key Result

Theorem 1

If $\mathcal{P}=(\Pi(\Delta^1),\Pi(\Delta^2))$ is an irreducible and quasismooth good pair of generalized nef partitions, then $g_1,\dots,g_s$ define a family of Calabi-Yau varieties $\mathcal{F}_{\mathcal{P}}$ in the ${\mathbb Q}$-Fano toric variety $Z_{\Delta^2}$.

Figures (4)

  • Figure 1: Generalized nef partition $\Pi(\Delta)$
  • Figure 2: Generalized nef partitions $\Pi(\Delta)$ and $\Pi(\nabla)$
  • Figure 3: Dual nef partitions $\Pi(\Delta)$ and $\Pi(\nabla)$
  • Figure 4: The pair $(\Delta^1,\Delta^2)$

Theorems & Definitions (80)

  • Theorem : see Theorem \ref{['CY']}
  • Proposition : see Proposition \ref{['prop-codim2']}
  • Lemma 1.1
  • proof
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Lemma 1.5
  • proof
  • Example 1.6
  • ...and 70 more