The tropicalisation of a $(-2,0)$-flop

TL;DR

This paper extends the combinatorial understanding of flops in algebraic geometry by constructing a cone in an affine manifold to describe a $(-2,0)$-flop, which cannot be realized torically, and explores mirror symmetry implications.

Contribution

It introduces a novel construction of an affine manifold cone to describe the $(-2,0)$-flop and establishes a mirror singularity, advancing the combinatorial approach in non-toric flops.

Findings

Constructed a cone in an affine manifold for the $(-2,0)$-flop

Described the flop via two subdivisions of the cone

Built a mirror singularity related to the original

Abstract

As a standard example in toric geometry, the Atiyah flop of a $(-1,-1)$-curve in a smooth 3-fold can be described combinatorially in terms of the two possible triangulations of a square cone. The flop of $(-2,0)$-curve cannot be realised in terms of toric geometry. Nevertheless, we explain how to construct a cone $\sigma$ in an integral affine manifold with singularities associated to the singularity $(P\in Y)$ at the base of a $(-2,0)$-flop. The two sides of the flop can then be described combinatorially in terms of two different subdivisions of $\sigma$. As an interesting byproduct of our construction, we can build a singularity $(P^\star\in Y^\star)$ which is mirror to $(P\in Y)$.