Quasi-period collapse for duals to Fano polygons: an explanation arising from algebraic geometry

2018/10/30 by Alexander Kasprzyk, Kasprzyk, Alexander M., Ben Wormleighton +1
Mathematics · #14J17 (Secondary) #52B20 (Primary) 05E #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1810.12472

openalex publication_date 2018/10/30 · openalex created_date 2018/11/09 · openalex updated_date 2026/07/28

Abstract

The Ehrhart quasi-polynomial of a rational polytope P is a fundamental invariant counting lattice points in integer dilates of P. The quasi-period of this quasi-polynomial divides the denominator of P but is not always equal to it: this situation is called quasi-period collapse. Polytopes experiencing quasi-period collapse appear widely across algebra and geometry, and yet the phenomenon remains largely mysterious. Using techniques from algebraic geometry - specifically the ℚ-Gorenstein deformation theory of orbifold del Pezzo surfaces - we explain quasi-period collapse for rational polygons dual to Fano polygons and describe explicitly the discrepancy between the quasi-period and the denominator.

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