Noncommutative deformations and flops
There are several invariants that can be derived from the flip or flop of a rational curve inside a 3-fold. In [DW1], finite-dimensional algebras (so-called noncommutative deformation algebras) are associated to these birational transformations, which unify the previously known invariants. Additionally, deformations over these algebras generalise classic deformations over (commutative) Artinian algebras.The aim of this seminar would be to understand the article [DW1], and maybe also a bit the subsequent article [W], where these ideas are applied to the bigger picture of a homological minimal model program.
Talks
The first session of this seminar took place on Saturday, January 16, 2016.- 10:30 – 12:00
David Ploog: Overview - 13:30 – 15:00
Wolfgang Ebeling: Singularities of threefolds and their resolutions
- 15:30 – 17:00
Stephen Coughlan: (Classical) deformation theory
- 10:00 – 11:15
Nathan Broomhead: Three-dimensional flops and non-commutative rings [vdB] - 11:30 – 12:45
Stephen Coughlan: Noncommutative deformations and flops [DW1, Sections 2 and 3] - 14:00 – 15:15
Andreas Hochenegger: Contractions and deformations [DW3]
- 10:00 – 11:15
Stephen Coughlan: On multi-pointed non-commutative deformations and Calabi-Yau threefolds [K] - 11:30 – 12:45
Andreas Krug: Mukai flops and P-twists [ADM] - 14:00 – 15:15
David Ploog: Contraction algebra and invariants of singularities [HT]
- 10:15 – 11:15
Lutz Hille: Flops and spherical functors, part 1 [BB] - 11:30 – 12:30
Lutz Hille: Flops and spherical functors, part 2 - 14:00 – 15:15
Greg Stevenson: DG-categories [S] - 15:45 – 17:00
David Ploog: Reminder on Three-dimensional flops and non-commutative rings [vdB]
If you want to be kept informed, please write an email.
Literature
[ADM] Mukai flops and P-twists by N. Addington, W. Donovan and C. Meachan. [BB] Flops and spherical functors by A. Bodzenta and A. Bondal. [DW1] Noncommutative deformations and flops by W. Donovan and M. Wemyss. [DW2] Twists and braids for general 3-fold flops by W. Donovan and M. Wemyss. [DW3] Contractions and deformations by W. Donavan and M. Wemyss. [HT] Contraction algebra and invariants of singularities by Z. Hua and Y. Toda. [K] On multi-pointed non-commutative deformations and Calabi-Yau threefolds by Y. Kawamata. [M] Minimal models of canonical 3-folds by M. Reid. [S] Six operations on dg enhancements of derived categories of sheaves by O. Schnürer.[vdB] Three-dimensional flops and non-commutative rings by M. van den Bergh. [W] Aspects of the Homological Minimal Model Program by M. Wemyss.