Melanie Wood : Discriminants in the Grothendieck Ring

Posted by Izzet Coskun , part of the Algebraic Geometry Seminar.

At

Feb. 20, 2013, 4 p.m.

In

SEO 427

Abstract

We consider the "limiting behavior" of *discriminants* (or their complements), by which we mean informally the closed locus in some parameter space of some type of object where the objects have certain singularities. We focus on the collection of unordered points on a variety X, and linear systems on X. These are connected --- we use the first to understand the second. We describe their classes in the Grothendieck ring of varieties, as the number of points gets large, or as the line bundle gets very positive. As applications, (i) we show the motivic analogue of Poonen's point-counting result: the motivic probability of a section of L being smooth (as L gets large) is 1 / Z_X( \A^{-\dim X - 1} ) (where Z_X is the motivic zeta functions), and (ii) show a priori unexpected structure in configuration spaces of points on a variety, leading to many topological and point-counting consequences and conjectures. This is joint work with Ravi Vakil.