Outline

In 1964, Hironaka proved that the singularities of any algebraic variety in characteristic 0 can be resolved by repeatedly blowing up suitably chosen smooth subvarieties, via a complicated algorithm. Since then, singularity theorists and birational geometers have been looking for a simpler algorithm, as well as for a better theoretical understanding of the method and its limitations.

In the late 2010s, it was discovered by Abramovich–Temkin–Włodarczyk [ATW] and McQuillan [McQ] that replacing blowups by stacky weighted blowups along smooth subvarieties allows for a much easier algorithm, where many of the previous technicalities disappear. The expense is that the output will be a smooth Deligne–Mumford stack, but replacing that by a smooth variety again is not very difficult. The completes a 50-year search for an easy and natural resolution algorithm, which can now be understood by a broad algebraic geometry audience.

The goal of this seminar is to present the proof of [ATW]. In the first session, we introduce the problem and outline the method. In the second, we explain some of the ingredients from the state of the art in 2007 [Kol], which still play a role in the new proof. The third session will go over the tools from algebraic stacks and new tools in [ATW], and the final session puts everything together and finishes the proof.

Organisers

The organisers of the seminar are:

Remy van Dobben de Bruyn	Universiteit Utrecht
Olivier de Gaay Fortman	Universiteit Utrecht
Simon Pepin Lehalleur	Universiteit van Amsterdam

If you would like to give a talk, please write an email to the organisers.

For practical questions about a meeting, please write to the local organisers at that university. In addition to the above, these are:

Universiteit Leiden	David Holmes
Radboud Universiteit Nijmegen	Simon Pepin Lehalleur

Meetings

We will meet at different universities in the country on 4 Fridays. There will be talks 13:00–14:00, 14:30–15:30, and 16:00–17:00.

• 21 February, Universiteit Utrecht.
• 7 March, *.
• 4 April, *.
• 9 May, *.

* Location to be confirmed.

References

The main reference is:

Other references:

Schedule

Click on the talks below to see a brief abstract (where available). The schedule below may be updated with further details, but should otherwise be stable.

21 February: Universiteit Utrecht

The first session is an introduction and overview of the seminar: what, why, and how?

Location: Buys Ballot building, room 161. Enter through the Victor J. Koningsberger building, go up the big stairs to the first floor, and keep walking straight towards the end of the first floor of Buys Ballot.

7 March: Location TBA

The second session is a more in-depth overview of Hironaka's method: introduce some key notions and explain the limitations of the method.

Lecture room TBA.

4 April: Location TBA

The third session introduces the new ingredients for Abramovich–Temkin–Włodarczyk's proof.

Lecture room TBA.

9 May: Location TBA

In the final session, we will complete Abramovich–Temkin–Włodarczyk's proof.

Lecture room TBA.