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A. 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 is called quasi-period collapse. Polytopes experiencing quasi-period collapse appear widely across algebra and geometry, and yet the phenomenon remains largely mysterious. By using techniques from algebraic geometry -Q-Gorenstein deformations 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. ILet P ⊂ Z d ⊗ Z Q be a convex lattice polytope of dimension d. Let L P (k) : kP ∩ Z d count the number of lattice points in dilations kP of P, k ∈ Z ≥0 . Ehrhart [9] showed that L P can be written as a degree dwhich we call the Ehrhart polynomial of P. The leading coefficient c d is given by Vol(P)/d!, c d−1 is equal to Vol(∂P)/2(d − 1)!, and c 0 1. Here Vol( • ) denotes the normalised volume, and ∂P denotes the boundary of P. For example, if P is two-dimensional (that is, P is a lattice polygon) we obtainSetting k 1 in this expression recovers Pick's Theorem [16]. The values of the Ehrhart polynomial of P form a generating function Ehr P (t) : k ≥0 L P (k)t k called the Ehrhart series of P. When the vertices of P are rational points the situation is more interesting. Recall that a quasi-polynomial with period s ∈ Z >0 is a function q : Z → Q defined by polynomials q 0 , q 1 , . . . , q s−1 such that q(k) q i (k) when k ≡ i (mod s).The degree of q is the largest degree of the q i . The minimum period of q is called the quasi-period, and necessarily divides any other period s. Ehrhart showed that L P is given by a quasi-polynomial of degree d, which we call the Ehrhart quasi-polynomial of P. Let π P denote the quasi-period of P. The smallest positive integer r P ∈ Z >0 such that r P P is a lattice polytope is called the denominator of P. It is certainly the case that L P is r P -periodic, however it is perhaps surprising that the quasi-period of L P does not always equal r P ; this phenomenon is called quasi-period collapse.Example 1.1 (Quasi-period collapse). Consider the triangle P : conv{(5, −1), (−1, −1), (−1, 1/2)} with denominator r P 2. This has L P (k) 9/2k 2 + 9/2k + 1, hence π P 1.Quasi-period collapse is poorly understood, although it occurs in many contexts. For example, de 8] consider polytopes arising naturally in the study of Lie algebras (the Gel'fand-Tsetlin polytopes and the polytopes determined by the Clebsch-Gordan coefficients) that exhibit quasiperiod collapse. In dimension two show that there exist rational polygons with r P arbitrarily large but with π P 1 (see also Example 3.8). give a constructive view of this phenomena in terms of GL d (Z)-scissor congruence; here a polytope is partitioned into pieces that are individually modified via GL d (Z) transformation and lattice translation, then reassembled to give a new polyt...

A. ECH capacities are rich obstructions to symplectic embeddings in 4-dimensions that have also been seen to arise in the context of algebraic positivity for (possibly singular) projective surfaces. We extend this connection to relate general convex toric domains on the symplectic side with towers of polarised toric surfaces on the algebraic side, and then use this perspective to show that the sub-leading asymptotics of ECH capacities for all convex and concave toric domains are p1q. We obtain sufficient criteria for when the sub-leading asymptotics converge in this context, generalising results of Hutchings and of the author, and derive new obstructions to embeddings between toric domains of the same volume. We also propose two invariants to more precisely describe when convergence occurs in the toric case. Our methods are largely non-toric in nature, and apply more widely to towers of polarised Looijenga pairs.