Smoothing toric Fano surfaces using the Gross-Siebert algorithm

Abstract

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 boldQ‐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.

Del Pezzo surfaces with $$\frac{1}{3}(1,1)$$ 1 3 ( 1 , 1 ) points

“…We put our results in the context of a general program to understand mirror symmetry for orbifold del Pezzo surfaces [2,16,26,31,27,10] and answer the question: Why classify del Pezzo surfaces with 1 3 (1, 1) points? qG-deformations of surface singularities is a technical notion that ensures that the canonical class is wellbehaved in families: in particular, K 2 is locally constant in a qG-family of proper surfaces.…”

“…This overlaps with work of Fujita and Yasutake [14].(II) Construct good models for surfaces in all families as degeneracy loci in rep quotient varieties. In all but two cases, the rep quotient variety is a simplicial toric variety.(III) Prove that precisely 26 of the 29 families admit a qG-degeneration to a toric surface.The classification is summarised in table 1 and table 2, which also plot invariants and provide good model constructions of surfaces in all families.This work is part of a program to understand mirror symmetry for orbifold del Pezzo surfaces [2,16,26,31,27,10] and it is aimed specifically at giving evidence for the conjectures made in [2].The rest of the introduction is organised as follows: in § 1.1 we give precise statements of our main results; in § 1.2 we say a few words about the context of [2]; in § 1.3 we outline the structure of the paper.…”

“…This work is part of a program to understand mirror symmetry for orbifold del Pezzo surfaces [2,16,26,31,27,10] and it is aimed specifically at giving evidence for the conjectures made in [2].…”

Minimality and Mutation-Equivalence Of polygons

“…Indeed, a mirror construction for log Calabi-Yau surfaces is described by Gross-Hacking-Keel in [21], and in [36] it is shown that, in this context, the corresponding family does indeed arise from the smoothing of the toric variety X P .…”

“…In particular, allowing singularities to collide with boundary points of the affine manifold creates corners and provides an analogue for the toric qG-degenerations of the surface. This process, and its lifting via the Gross-Siebert program to construct the corresponding degeneration of algebraic varieties, are described in [36].…”