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

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 $(P\in Y)$.