Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 095, 20 pages      arXiv:2112.08246      https://doi.org/10.3842/SIGMA.2024.095

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

Wendelin Lutz
Department of Mathematics and Statistics, Lederle Graduate Research Tower, University of Massachusetts, Amherst, MA 01003-9305, USA

Received May 07, 2024, in final form October 14, 2024; Published online October 22, 2024

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 $X$.

Key words: $T$-polygons; mirror symmetry; del Pezzo surfaces; mutations; maximally mutable Laurent polynomial.

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