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Geometry of tropical mutation surfaces with a single mutation

Tomoki Oda

Abstract

Recently, Escobar, Harada, and Manon introduced the theory of polyptych lattices. This theory gives a general framework for constructing projective varieties from polytopes in a polyptych lattice. When all the mutations of the polyptych lattice are linear isomorphisms, this framework recovers the classical theory of toric varieties. In this article, we study rank two polyptych lattices with a single mutation. We prove that the associated projective surface $X$ is a $\mathbb{G}_m$-surface that admits an equivariant $1$-complement $B\in |-K_X|$ such that $B$ supports an effective ample divisor. Conversely, we show that a $\mathbb{G}_m$-surface $X$ that admits an equivariant $1$-complement $B\in |-K_X|$ supporting an effective ample divisor comes from a polyptych lattice polytope. Finally, we compute the complexity of the pair $(X,B)$ in terms of the data of the polyptych lattice, we describe the Cox ring of $X$, and study its toric degenerations.

Geometry of tropical mutation surfaces with a single mutation

Abstract

Recently, Escobar, Harada, and Manon introduced the theory of polyptych lattices. This theory gives a general framework for constructing projective varieties from polytopes in a polyptych lattice. When all the mutations of the polyptych lattice are linear isomorphisms, this framework recovers the classical theory of toric varieties. In this article, we study rank two polyptych lattices with a single mutation. We prove that the associated projective surface is a -surface that admits an equivariant -complement such that supports an effective ample divisor. Conversely, we show that a -surface that admits an equivariant -complement supporting an effective ample divisor comes from a polyptych lattice polytope. Finally, we compute the complexity of the pair in terms of the data of the polyptych lattice, we describe the Cox ring of , and study its toric degenerations.

Paper Structure

This paper contains 8 sections, 38 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classification of affine tropical mutation varieties associated with Ms
  4. Singularities of tropical mutation varieties
  5. The complexity of tropical mutation surface pairs
  6. Cluster type surfaces as tropical mutation surfaces
  7. Toric degenerations of tropical mutation varieties
  8. Cox rings of tropical mutation surfaces

Key Result

Theorem 1.1

Let $s\ge 1$. The coarse moduli space of detropicalizations of the polyptych lattice $\mathcal{M}_s$ is given by the quotient where $D_{2s}$ denotes the dihedral group generated by the cyclic subgroup $\mu_s$ of $s$-th roots of unity, generated by $\zeta$ and a reflection $s$. The group $D_{2s}$ acts on the coordinate space $\mathbb{A}^{\,s-1}=\operatorname{Spec}\mathbb{K} [b_1,\dots,b_{s-1}]$ by

Theorems & Definitions (83)

  • Theorem 1.1: cf. Theorem \ref{['thm:Ms-moduli']}
  • Theorem 1.2: cf. Corollary \ref{['complexity']}
  • Theorem 1.3: cf. Theorem \ref{['thm:cluster-tropical']}
  • Theorem 1.4: cf. Theorem \ref{['toricdegene']}, Theorem \ref{['coincidewith']}
  • Theorem 1.5: cf. Theorem \ref{['X_s Cox']}, Corollary \ref{['criteria']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • ...and 73 more