# 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.

## 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).

## Spring 2015 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 | |||

March 6 | Matt Satriano (Johns Hopkins) | When is a variety a quotient of a smooth variety by a finite group? | Max |

March 13 | Jose Rodriguez (Notre Dame) | Numerical irreducible decomposition of multiprojective varieties | Daniel |

March 20 | Dima Arinkin (Wisconsin) | Smooth categorical representations of reductive groups | |

March 27 | Joerg Schuermann (Muenster) | Chern classes and transversality for singular spaces | Max |

April 17 | Lee McEwan (OSU, Mannsfield) | Topology of Quasi-Ordinary Surfaces | Max and Gonzalez Villa |

April 24 | Matthew Woolf (UIC) | TBA | Daniel |

May 1 | Byeongho Lee (Purdue) | $G$-Frobenius manifolds | Andrei |

May 8 | Brian Lehmann (Boston College) | TBA | Daniel |

## 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.

### Matt Satriano

When is a variety a quotient of a smooth variety by a finite group?

We explore the following local-global question: if X is locally a quotient of a smooth variety by a finite group, then is it globally of this form? We show that the answer is "yes" whenever X is quasi-projective and already known to be a quotient by a torus. In particular, this applies to all quasi-projective simplicial toric varieties. We discuss the proof and show how it can be made explicit in the case of toric varieties. This is joint work with Anton Geraschenko.

### Jose Rodriguez

Numerical algebraic geometry is a growing area of algebraic geometry that involves describing solution sets of systems of polynomial equations. This area has already had an impact in kinematics, statistics, PDE's, and pure math. This talk will introduce key concepts in numerical algebraic geometry that are used to describe positive dimensional projective varieties. In particular, witness sets will be defined and the classic "regeneration procedure" will be described. The second part of the talk will describe a new "Multi-Regeneration Procedure". This technique gives an effective way of describing multiprojective varieties and determining their multidegrees.

Throughout the talk motivating examples will be provided, and no previous knowledge of numerical algebraic geometry will be assumed. This is joint work with Jonathan Hauenstein.

### Dima Arinkin

Smooth categorical representations of reductive groups

Let G be a complex reductive group. Consider categories equipped with smooth (sometimes called strong) action of G. Natural and important examples of such categories arise from geometry: if X is a variety equipped with an action of G, for instance, X=G itself, or X=G/B is the flag space of G, then the category of D-modules on X carries a smooth action of G. We view such categories as (smooth) categorical representations of G.

The theory of smooth categorical representations of G is similar to the representation theory of a reductive group over a finite field, I will discuss this similarity in my talk. The surprising twist (and the main result of this talk) is that the theory of smooth categorical representations is simpler than its classical counterpart: there are no cuspidal representations!

### 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.

### Lee McEwan

Topology of Quasi-Ordinary Surfaces

Let [math]X = { f(z; x_1, \dots x_n) = 0 }[/math] be a reduced hypersurface germ in [math](\mathbb{C}^{n+1}, 0)[/math] and suppose [math]\pi : X \rightarrow (\mathbb{C}^n, 0)[/math] is a finite projection with discriminant a normal crossings divisor. Such an [math]X[/math] is quasi-ordinary, generalizing the idea of ordinary singularities. This talk will review the recent history in the topological study of such objects and their smoothings. Then we will describe current research on the Milnor fiber and its boundary in the case [math]n = 2[/math].

### Byeongho Lee

$G$-Frobenius manifolds

The goal of this talk is to introduce the problem of orbifolding Frobenius manifolds and a related concept of $G$-Frobenius manifolds for each finite group $G$. Frobenius manifolds are among the central players of classical mirror symmetry, and orbifolding them can be described as producing a new Frobenius manifold when the original one has a certain group symmetry. After giving some background, $G$-Frobenius manifolds will be introduced as an ingredient of the procedure of orbifolding.