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.
Algebraic Geometry Mailing List
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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 [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] intersect transversal in a Whitney stratified sense, this is true for the MacPherson Chern classes (of adopted constructible functions). If [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] 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 [math]\displaystyle{ M }[/math] with respect to [math]\displaystyle{ X\times Y }[/math]. This notion of non-characteristic is weaker than the Whitney stratified transversality as well as the splayedness assumption.