Sue Sierra, University of Edinburgh
“Birational transformations of noncommutative projective surfaces”
Noncommutative algebraic geometry approaches noncommutative rings through geometric techniques. This has has the most success in studying noncommutative graded rings. A graded domain of dimension 3 can be considered to be (the coordinate ring of) a noncommutative projective surface. What are the noncommutative projective surfaces?
In classical algebraic geometry, surfaces are studied first by describing them up to birational equivalence and then by working within a birational classification to describe the ”minimal models” within each class. In noncommutative geometry, the question of birational equivalence is famously difficult, but there has been much progress from working within a birational class. We give an overview of the theory of noncommutative projective surfaces, and describe recent work (joint with Rogalski and Stafford) on a noncommutative version of Castelnuovo’s celebrated contraction criterion: a line of self-intersection (-1) on a smooth surface can be contracted to a point.
Refreshments will be served in MC 5403 at 3:30 p.m. Everyone is welcome to attend.