Mirror Symmetry for log Calabi-Yau Surfaces II

We show that the ring of regular functions of every smooth affine log Calabi-Yau surface with maximal boundary has a vector space basis parametrized by its set of integer tropical points and a $\mathbb{C}$-algebra structure with structure coefficients given by the geometric construction of Keel-Yu. To prove this result, we first give a canonical compactification of the mirror family associated with a pair $(Y,D)$ constructed by Gross-Hacking-Keel where $Y$ is a smooth projective rational surface, $D$ is an anti-canonical cycle of rational curves and $Y\setminus D$ is the minimal resolution of an affine surface with, at worst, du Val singularities. Then, we compute periods for the compactified family using techniques from work of Ruddat-Siebert and use this to give a modular interpretation of the compactified mirror family.