Mirrors to Del Pezzo Surfaces and the Classification of $T$-Polygons

TL;DR

This paper provides a geometric proof of the classification of T-polygons using mirror symmetry, showing that toric degenerations of del Pezzo surfaces are connected through rational curves in the moduli space.

Contribution

It offers a new geometric approach to classifying T-polygons and demonstrates the connectedness of toric degenerations of del Pezzo surfaces via mirror symmetry techniques.

Findings

Geometric proof of T-polygons classification

Connectedness of toric degenerations in moduli space

Application of mirror symmetry to algebraic geometry

Abstract

We give a new geometric proof of the classification of $T$-polygons, a theorem originally due to Kasprzyk, Nill and Prince, using ideas from mirror symmetry. In particular, this gives a completely geometric proof that any two toric $\mathbb{Q}$-Gorenstein degenerations of a smooth del Pezzo $X$ surface are connected via trees of rational curves in the moduli space of $X$.