Algebraic Geometry Seminar Spring 2015

The seminar meets on Fridays at 2:25 pm in Van Vleck B135.

The schedule for the previous semester is here.

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Fall 2014 Schedule

date	speaker	title	host(s)
January 30	Manuel Gonzalez Villa (Wisconsin)	Motivic infinite cyclic covers
February 20	Jordan Ellenberg (Wisconsin)	Furstenberg sets and Furstenberg schemes over finite fields	I invited myself
February 27	Botong Wang (Notre Dame)	TBD	Max
March 6	Matt Satriano (Johns Hopkins)	TBD	Max
March 13	Jose Rodriguez (Notre Dame)	TBD	Daniel
March 27	Joerg Schuermann (Muenster)	Chern classes and transversality for singular spaces	Max

Abstracts

Manuel Gonzalez Villa

Motivic infinite cyclic covers (joint work with Anatoly Libgober and Laurentiu Maxim)

We associate with an infinite cyclic cover of a punctured neighborhood of a simple normal crossing divisor on a complex quasi-projective manifold (assuming certain finiteness conditions are satisfied) an element in the Grothendieck ring, which we call motivic infinite cyclic cover, and show its birational invariance. Our construction provides a unifying approach for the Denef-Loeser motivic Milnor fibre of a complex hypersurface singularity germ, and the motivic Milnor fiber of a rational function, respectively.

Jordan Ellenberg

Furstenberg sets and Furstenberg schemes over finite fields (joint work with Daniel Erman)

We prove a theorem of Kakeya type for the intersection of subsets of n-space over a finite field with k-planes. Let S be a subset of F_q^n with the "k-plane Furstenberg property": for every k-plane V, there is a k-plane W parallel to V which intersects S in at least q^c points. We prove that such a set has size at least a constant multiple of q^{cn/k}. The novelty is the method; we prove that the theorem holds, not only for subsets of the plane, but arbitrary 0-dimensional subschemes, and reduce the problem by Grobner methods to a simpler one about G_m-invariant subschemes supported at a point. The talk will not assume that everyone in the room is an algebraic geometer.

Jose Rodriguez

TBA

Joerg Schuermann

Chern classes and transversality for singular spaces

Let [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] be closed complex subvarieties in an ambient complex manifold [math]\displaystyle{ M }[/math]. We will explain the intersection formula [math]\displaystyle{ $$c(X) \cdot c(Y)= c(TM)\cap c(X\cap Y)$$ }[/math] for suitable notions of Chern classes and transversality for singular spaces. If $X$ and $Y$ intersect transversal in a Whitney stratified sense, this is true for the MacPherson Chern classes (of adopted constructible functions). If $X$ and $Y$ are "splayed" in the sense of Aluffi-Faber, then this formula holds for the Fulton-(Johnson-) Chern classes, and is conjectured for the MacPherson Chern classes. We explain, that the version for the MacPherson Chern classes is true under a micro-local "non-characteristic" condition for the diagonal embedding of $M$ with respect to $X\times Y$. This notion of non-characteristic is weaker than the Whitney stratified transversality as well as the splayedness assumption.