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arXiv · 1810.12472

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

Alexander M. Kasprzyk·Ben Wormleighton
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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 $\mathbb{Q}$-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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Alexander M. Kasprzyk, Ben Wormleighton. 2018-10-30. Quasi-period collapse for duals to Fano polygons: an explanation arising from algebraic geometry. https://arxiv.org/abs/1810.12472

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