A game for the resolution of singularities

Abstract

We propose a combinatorial game on finite graphs, called Stratify, that is played by two protagonists, 𝒜 and ℬ. The game captures the logical structure of a proof of the resolution of singularities. In each round, the graph of the game is modified by the moves of the players. When it assumes a final configuration, 𝒜 has won. Otherwise, the game goes on forever, and nobody wins. In particular, ℬ cannot win the game, but can only prevent 𝒜 from winning. We show that 𝒜 always possesses a winning strategy, regardless of the initial shape of the graph and of the moves of ℬ. This implies, translating the game back to algebraic geometry, that there is a choice of centres for the blowup of singular varieties in characteristic zero which eventually leads to their resolution. The algebra needed for this implication is elementary. The transcription from varieties to graphs and from blowups to modifications of the graph thus axiomatizes the proof of the resolution of singularities. In principle, the same logic could also work in positive characteristic, once an appropriate descent in dimension is settled.