The birational geometry of noncommutative surfaces

Abstract: In commutative algebraic geometry, the theory of smooth projective surfaces is, of course, very highly developed, with a major result being the birational classification of such surfaces. For the noncommutative analogue, much less is known, with even the notion of "birational" not being very well understood. In particular, although several constructions have been known (noncommutative projective planes, noncommutative ruled surfaces, and noncommutative blowups), many basic isomorphisms have proved elusive (e.g., that blowups in distinct points commute). I'll discuss a new approach to the problem via derived categories that not only makes it easy to construct the desired isomorphisms but also to prove a number of other results, in particular that anything birational to a ruled surface is either ruled or a projective plane, and the corresponding moduli spaces of simple sheaves are Poisson, with smooth symplectic leaves.

algebraic geometrydifferential geometrygeometric topologysymplectic geometry

Audience: researchers in the topic

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