Smoothing Toric Fano Surfaces Using the Gross-Siebert Algorithm

TL;DR

This paper develops a tropical and affine geometric approach to partially smoothing toric Fano surfaces with cyclic quotient singularities, advancing mirror symmetry understanding through a generalized Gross-Siebert algorithm.

Contribution

It introduces a generalized Gross-Siebert reconstruction method to construct partial smoothings from affine manifolds with singularities, extending previous rigidity assumptions.

Findings

Constructed affine manifolds with singularities for partial smoothing

Implemented a generalized Gross-Siebert algorithm

Linked partial smoothings to mirror symmetry expectations

Abstract

A toric del Pezzo surface $X_P$ 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 a tropical version of a Q-Gorenstein partial smoothing of $X_P$. 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.