Geometry of tropical mutation surfaces with a single mutation

Abstract

Escobar, Harada, and Manon introduced polyptych lattices as a piecewise-linear extension of the lattice-polytope formalism of toric geometry. In this paper we study the first genuinely non-toric case: rank-two polyptych lattices with a single shear. A detropicalization is given by a polynomial \(f(y)\), and the corresponding affine surface is $U_f=\operatorname{Spec} K[x_1,x_2,y^{\pm 1}]/\langle x_1x_2-f(y)\rangle.$ We classify these detropicalizations, compute the complexity of their projective compactifications, and show that the resulting log Calabi--Yau surface pairs are of cluster type. Conversely, we prove that every normal projective \(\mathbb Q\)-factorial index-one log Calabi--Yau surface pair with reduced boundary, ample boundary support, and a nontrivial \(\mathbb G_m\)-action arises from this single-shear construction. We also construct a global family interpolating between the two toric degenerations associated with the two charts, and compute the Cox rings of the resulting tropical mutation surfaces.