Gorenstein toric Fano varieties

2004/05/24 by Benjamin Nill · 9 citations
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #math.AG #math.CO #msc:14J45 #msc:14M25 #msc:52B20

paper · pdf · doi:10.1007/s00229-004-0532-3

published as Manuscripta Math. 116 (2005), 183-210 · AMS-LaTeX, 29 pages with 5 figures

arxiv created 2004/05/24 · openalex publication_date 2005/01/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29

Abstract

We investigate Gorenstein toric Fano varieties by combinatorial methods using the notion of a reflexive polytope which appeared in connection to mirror symmetry. The paper contains generalisations of tools and previously known results for nonsingular toric Fano varieties. As applications we obtain new classification results, bounds of invariants and formulate conjectures concerning combinatorial and geometrical properties of reflexive polytopes.

Citations

• Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties
• REFLEXIVE POLYHEDRA, WEIGHTS AND TORIC CALABI-YAU FIBRATIONS

Cited by

• The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning
• Toric Fano manifolds with nef tangent bundles
• Mirror Transitions and the Batyrev-Borisov construction
• The number of vertices of a Fano polytope
• Notes on the roots of Ehrhart polynomials
• Reflexive polytopes and the Picard ranks of Gorenstein toric Fano varieties
• Small toric degenerations of Fano threefolds
• Fano horospherical variety
• The reflexive dimension of a lattice polytope

Related

• Gorenstein Fano toric degenerations
• Q-factorial Gorenstein toric Fano varieties with large Picard number
• On smooth Gorenstein polytopes
• Reflexive polytopes and the Picard ranks of Gorenstein toric Fano varieties
• Towards the Mirror Symmetry for Calabi-Yau Complete intersections in Gorenstein Toric Fano Varieties