# Difference between revisions of "Algebraic Geometry Seminar Spring 2015"

(→Joerg Schuermann) |
(→Joerg Schuermann) |
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Line 70: | Line 70: | ||

Let <math>X</math> and <math>Y</math> be closed complex subvarieties in an ambient | Let <math>X</math> and <math>Y</math> be closed complex subvarieties in an ambient | ||

complex manifold <math>M</math>. We will explain the intersection formula | complex manifold <math>M</math>. We will explain the intersection formula | ||

− | <math> | + | <math>c(X) \cdot c(Y)= c(TM)\cap c(X\cap Y)</math> |

for suitable notions of Chern classes and transversality for singular spaces. | for suitable notions of Chern classes and transversality for singular spaces. | ||

− | If | + | If <math>X</math> and <math>Y</math> intersect transversal in a Whitney stratified sense, this is |

true for the MacPherson Chern classes (of adopted constructible functions). | true for the MacPherson Chern classes (of adopted constructible functions). | ||

− | If | + | If <math>X</math> and <math>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 | holds for the Fulton-(Johnson-) Chern classes, and is conjectured for the | ||

MacPherson Chern classes. | MacPherson Chern classes. | ||

We explain, that the version for the MacPherson Chern classes is true under a | We explain, that the version for the MacPherson Chern classes is true under a | ||

− | micro-local "non-characteristic" condition for the diagonal embedding of | + | micro-local "non-characteristic" condition for the diagonal embedding of <math>M</math> |

− | with respect to | + | with respect to <math>X\times Y</math>. This notion of non-characteristic is weaker than |

the Whitney stratified transversality as well as the splayedness assumption. | the Whitney stratified transversality as well as the splayedness assumption. |

## Revision as of 23:53, 30 January 2015

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

The schedule for the previous semester is here.

## Contents

## Algebraic Geometry Mailing List

- Please join the Algebraic Geometry Mailing list to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).

## 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]X[/math] and [math]Y[/math] be closed complex subvarieties in an ambient complex manifold [math]M[/math]. We will explain the intersection formula [math]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]X[/math] and [math]Y[/math] intersect transversal in a Whitney stratified sense, this is true for the MacPherson Chern classes (of adopted constructible functions). If [math]X[/math] and [math]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]M[/math] with respect to [math]X\times Y[/math]. This notion of non-characteristic is weaker than the Whitney stratified transversality as well as the splayedness assumption.