Quicklists
Javascript must be enabled

Melanie Matchett Wood : Motivic Discriminants

root

115 Views

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 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 function), and (ii) show a priori unexpected structure in configuration spaces of points on a variety, with topological and point-counting consequences. Some low-tech examples: if v is a partition of n \leq 9, and v \neq (1,1,2,2,3), then the v-discriminant in the space of degree n polynomials (those polynomials with those root multiplicities, or worse) can be cut-and-pasted into affine space. (Question: over \C, does the complement have only two nonvanishing cohomology groups? What structure remains when n is larger?) This is joint work with Ravi Vakil.

Please select playlist name from following

Report Video

Please select the category that most closely reflects your concern about the video, so that we can review it and determine whether it violates our Community Guidelines or isn’t appropriate for all viewers. Abusing this feature is also a violation of the Community Guidelines, so don’t do it.

0 Comments

Comments Disabled For This Video