Geometry and Topology Seminar - David Stapleton - Hypersurfaces which are far from being rational
Title:Hypersurfaces which are farfrom being rational
Abstract: Rational varieties aresome of the simplest examples of varieties, e.g. most of their points can beparametrized by affine space. It is natural to ask (1) How can we determinewhen a variety is rational? and (2) If a variety is not rational, can wemeasure how far it is from being rational? A famous particular case of thisproblem is when the variety is a smooth hypersurface in projectivespace. This problem has attracted a great deal of attention bothclassically and recently. The interesting case is when the degree of thehypersurface is at most the dimension of the projective space (the ``Fano"range) as these hypersurfaces share many of the properties of projective space.In this talk, we present a recent result, joint with Nathan Chen, which saysthat smooth Fano hypersurfaces of large dimension can have arbitrarily largedegrees of irrationality, i.e. they can be arbitrarily far from being rational.