UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

March 06, 2012

939 Evans Hall

3:45PM: Introduction to the Landscape of Generalized Euler Characteristics

Daniel Litt

This talk will be a gentle introduction to the Grothendieck ring of varieties (the "baby ring of motives") and its role in the search for computable invariants of varieties. This ring is important, mysterious, and in some ways pathological. I will describe various questions about the Grothendieck ring, motivated by arithmetic, topology, and geometry--in particular, an analogue of part of the Weil conjectures--as well as some of its pathologies. I'll answer some of these questions for curves and discuss how these answers manifest themes from arithmetic and topology. Time permitting, I'll discuss obstructions to the truth of related statements for surfaces. This talk should be a good introduction to the material in Ravi Vakil's talk during the following hour.

5:00PM: Stabilization of discriminants in the Grothendieck ring

Ravi Vakil

We consider the ``limiting behavior'' of {\em 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 points on a variety, 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 $\mathbb{C}$ 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.

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