Algebraic Geometry Seminar

Let X be a k-hypersurface of degree d. Assume that X(k′) is nonempty for some finite extension k′/k of degree coprime to d. When d=2, this condition implies that X(k) is nonempty. For d>3, the same statement is no longer true in general, but for d=3 it is conjectured to remain valid.

More generally, one may ask whether it is possible to bound the degree of the smallest field extension coprime to d over which X admits a point. In this talk, we interpret these questions in terms of the geometry of the symmetric powers Sym^n(X)​ and study the relationship between their stable birational types. In particular, we show that if X is a geometrically rational surface (for example, a cubic surface), then only finitely many stable birational classes occur among the Sym^n(X)​. As a corollary, we deduce the rationality of the motivic zeta function of X (with coefficients in K0​(Vark​)/[A1] and char(k)=0).