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Geometry and Topology Seminar - Brooke Ullery - Measures of irrationality for algebraic varieties


HH 102

Title: Measures of irrationality foralgebraic varieties

Speaker: Brooke Ullery, Harvard

Abstract: A smooth algebraic curve is said to be rational if it isisomorphic to P^1, the projective line. More generally, the gonality of asmooth projective curve is the smallest degree of a map from the curve to theprojective line. The intuition is that the higher the gonality, the further thecurve is from being rational. A classical theorem of Noether says that if C isa smooth plane curve of degree d, then the gonality of C is d-1, and itis obtained by projecting away from a point on the curve. A natural questionis: does Noether's theorem generalize in some way to curves in largerprojective spaces? What about to higher dimensional varieties? We will explorethese questions, focusing on the examples of hypersurfaces and, more generally,complete intersections in projective space.
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