Colloquium: "How complicated could a Fano hypersurface really be?"
Abstract: The solutions set of a homogeneous polynomial is the most basic and well-studied example of an algebraic variety. It is interesting to ask -- how complicated can these varieties be? Riemann introduced the notion of the 'gonality' of a compact Riemann surface, which is a natural number that gives one possible answer to this question (e.g. the gonality of X is 1 if and only if X is the Riemann sphere). In higher dimensions the gonality is generalized by the degree of irrationality, denoted irr(X). When the degree d of a hypersurface is large compared to the number of variables n, it is relatively easy to show that irr(X) is approximately d. However, when d<n (the 'Fano range'), it is a famously difficult problem to show that irr(X) is not equal to 1. In this talk we discuss how reduction modulo p can be used to show that Fano hypersurfaces have arbitrarily large degrees of irrationality.
Host: Roya Beheshti