A game for the resolution of singularities

Abstract

We propose a combinatorial game on finite graphs, called Salmagundy, that is played by two protagonists, Dido and Mephisto. 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, Dido has won. Otherwise, the game goes on forever, and nobody wins. In particular, Mephisto cannot win himself, he can only prevent Dido from winning.We show that Dido always possesses a winning strategy, regardless of the initial shape of the graph and of the moves of Mephisto. This implies -translating back to algebraic geometry -that there is a choice of centers 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.

Handbook of Geometry and Topology of Singularities I

“…Much work has been done since 1964 to simplify and better understand resolution of singularities in characteristic zero. We mention [34], [19]- [22], [24]- [26], [29]- [33], [41]- [42], [64], [75], [76], [89], [153]- [155], [163], [166], [169] and [170].…”

Alternative invariants for the embedded resolution of purely inseparable surface singularities

“…A more axiomatic approach to resolution has been proposed by Hauser and Schicho [25]: The various specific constructions of the classical proof in characteristic zero are replaced by their key properties. These in turn suffice to give a purely combinatorial description of the entire resolution argument in form of a game (a viewpoint which orignally goes back to Hironaka).…”

Blurred combinatorics in resolution of singularities: (A little) Beyond the characteristic polytope

“…Indeed, it is possible to see transformations of the ideal as modifications of the polytope. Hironaka himself turned these ideas into his so-called polyhedra games, which were solved by Spivakovsky [11,12] (see [4] for a different, modern take).…”