# Algebraic Geometry Seminar Spring 2015: Difference between revisions

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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 04:53, 31 January 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

- 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]\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.