Smoothing toric Fano surfaces using the Gross-Siebert algorithm

A toric del Pezzo surface XP with cyclic quotient singularities determines and is determined by a Fano polygon P.We construct an affine manifold with singularities that partially smooths the boundary of P; this is a tropical version of a Q-Gorenstein partial smoothing of XP . We implement a mild generalization of the Gross–Siebert reconstruction algorithm – allowing singularities that are not locally rigid – and thereby construct (a formal version of) this partial smoothing directly from the affine manifold. This has implications for mirror symmetry: roughly speaking, it implements half of the expected mirror correspondence between del Pezzo surfaces with cyclic quotient singularities and Laurent polynomials in two variables.