Hironaka’s resolution of singularities

The October Notices of the AMS is already out. It features an interview with Heisuke Hironaka. Hironaka is most famous for his proof of the existence of a resolution of singularities for an algebraic variety: every algebraic variety is birationally equivalent to a smooth variety, and the birational equivalence can be realized as a sequence of blow ups. The proof involves a famously fiendish sextuple induction. For a nice introduction, take a look at Hauser’s article, Hironaka Theorem on Resolution of Singularities.

Hironaka’s proof only works in characteristic zero, so a major research problem has been the situation in characteristic p. Abhyankar has proven it in the case of surfaces, but as far as I know, the question is still open in higher dimensions. Interestingly, people have been able to prove weaker results but by going in a radically different direction. The review of the book Alterations and resolution of singularities from the Bulletin provides some details.

Peter Woit spotted the new issue of the Notices a couple of days ago, and has some comments on the contents. He also passes along the interesting fact that Hironaka is celebrity in Japan, a big enough one that he appears on billboards.

One thought on “Hironaka’s resolution of singularities

  1. just heard recently that resolution in char p is solved, hironaka again!, he’s giving a lecture about it in the ongoing confernce on resolution of singularity at ICTP, Trieste.

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