|Speaker:||Will Donovan (Kavli IPMU)|
|Title:||Geometry of 3-folds, and noncommutative deformations|
|Date (JST):||Mon, Dec 08, 2014, 14:00 - 17:00|
|Place:||Seminar Room A|
Rational curves on complex 3-folds are a rich source of geometric interest. I will discuss a new approach to their birational and enumerative geometry, which focuses on their noncommutative deformations. This is joint work with M. Wemyss. Under suitable geometric assumptions, we associate an associative algebra of deformations to each curve, and use this algebra to control birational modifications of the 3-fold, and investigate the structure of its derived category. The algebra may be viewed as an invariant of the curve, and has a role in unifying and generalising other classical invariants.
In the first part, I will give a motivation for the use of noncommutative methods in algebraic geometry, introduce some of the relevant geometry through examples, and summarize our results. In the second part, I will explain some key proofs, go deeper into the homological algebra motivating our constructions, and speak about work in progress.