Ravi Vakil

Abstract: We consider the ``limiting behavior'' of discriminants, by which we mean informally the closure of the locus in some parameter space of some type of object where the objects have certain singularities. Examples include configuration spaces of points on a variety (such as a complex projective manifold), and hypersurfaces on a variety. The classes of the discriminants in the "ring of motives" tend to a limit (in some appropriate sense) as the number of points (or the degree of the hypersurface) goes to infinity. (This ring is not scary, and I will define everything from scratch.) The "limits" have remarkable structure, in terms of "motivic zeta values" (which will be defined in the talk). The results parallel a number of results in both arithmetic and topology, and suggest a number of conjectures in both. (Two motivating classical facts: the chance of an integer being square free is 1 / zeta(2). The space of polynomials over C with no triple root has two non-zero rational cohomology groups: h^0=1 and h^3=1.) I will also present a conjecture (motivated by results in arithmetic and topology) suggested by our work. Although it is true in important cases, Daniel Litt has shown that it contradicts other hoped-for statements. This is joint work with Melanie Wood.