Algebraic Geometry Seminar
Melanie Wood
University of Wisconsin
Discriminants in the Grothendieck Ring
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.
Wednesday February 20, 2013 at 4:00 PM in SEO 427