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Abstracts for Online reading group "Intersection theory on stacks"

Alternatively have a look at the program.

Online: Category background for stacks

Posted in
Speaker: 
Reinier Kramer
Affiliation: 
MPIM
Date: 
Mon, 2020-07-06 10:30 - 12:00

I will recall often-used categorical constructions, such as the Yoneda lemma, categorical limits, and adjunctions. Most examples will be algebraic or topological in nature, with more geometric examples coming in the next session. I will also introduce group objects and discrete group objects in a category.

 

Online: Recap of scheme theory

Posted in
Speaker: 
Reinier Kramer
Affiliation: 
MPIM
Date: 
Thu, 2020-07-09 13:30 - 15:00

I will recall the definition of sheaves and schemes and many of their properties, such as e.g. properness, smoothness, &c. This is all material from Hartshorne, parts II and III, with less of a focus on sheaf cohomology (already treated in the previous reading group, on DT invariants), and making use of the category theory background.

 

Sites and sheaves

Posted in
Speaker: 
Reinier Kramer
Affiliation: 
MPIM
Date: 
Mon, 2020-07-13 10:30 - 12:00

I will define sites, i.e. categories with a Grothendieck topology on them. I will give several examples of sites of topological spaces and of schemes. Sites are the right categorical context for sheaf theory, and I will explain how. Finally, I will sketch a proof of Grothendieck's result that representable functors are sheaves in the fpqc topology - and hence also in the fppf and étale topology. This is mostly based on Vistoli's notes, section 2.3.

 

Fibered categories, I

Posted in
Speaker: 
Praphulla Koushik
Affiliation: 
School of Mathematics, IISER Thiruvananthapuram, India
Date: 
Mon, 2020-07-20 10:30 - 12:00

In this talk, I will first introduce the notions of fibered categories over a category, pseudo-functors over a category and then give a correspondence between "fibered categories over a category C" and pseudo-functors over C. I will then give examples of fibered categories, in particular, the example of fibered category of quasi-coherent sheaves on Sch/S. I will then talk about special type of fibered categories, namely categories fibered in groupoids and categories fibered in sets. This is based on sections 3.1-3.4 of Vistoli's notes.

 

Fibered categories (Part 2/2)

Posted in
Speaker: 
Alessandro Giacchetto
Affiliation: 
MPIM
Date: 
Mon, 2020-07-27 10:30 - 12:00

I will briefly recall the notion of fibered categories, and give the illustrative example of elliptic curves. I will then present some important results (foremost is Yoneda’s lemma for fibered categories), and conclude with a discussion of equivariant objects in a fibered category.


https://bbb.mpim-bonn.mpg.de/b/rei-xh2-kg6
For password email to rkramer@mpim...

Definition and examples of stacks

Posted in
Speaker: 
Campbell Wheeler
Affiliation: 
MPIM
Date: 
Mon, 2020-08-03 10:30 - 12:00

As motivation, we will start by considering the category of continuous functions and illustrate some gluing properties that make it a stack over the category of topological spaces. We will then give the definition of a stack and explore some other examples of stacks coming from algebraic geometry.

 

https://bbb.mpim-bonn.mpg.de/b/rei-xh2-kg6
For password email to rkramer@mpim...

Algebraic spaces and Algebraic stacks

Posted in
Speaker: 
Praphulla Koushik
Affiliation: 
School of Mathematics, IISER Thiruvananthapuram, India
Date: 
Mon, 2020-08-10 10:30 - 12:00

In this talk, after recalling the notion of a stack over an arbitrary site, I will introduce the notions of an algebraic space, an algebraic stack and a Deligne-Mumford stack. I will then introduce some examples and properties of these "generalized schemes".

 

https://bbb.mpim-bonn.mpg.de/b/rei-xh2-kg6
For password email to rkramer@mpim...

Moduli stacks of curves

Posted in
Speaker: 
Séverin Charbonnier
Affiliation: 
MPIM
Date: 
Mon, 2020-08-17 10:30 - 12:00

Using the machinery exposed in the previous talks, we will discuss the example of the moduli stacks of stable curves. We will show that it is an algebraic Deligne-Mumford stack and review some of its properties (irreducibility, properness). Last, we will briefly discuss the moduli of stable maps.

https://bbb.mpim-bonn.mpg.de/b/rei-xh2-kg6
For password email to rkramer@mpim...

Algebraic cycles and rational equivalence

Posted in
Speaker: 
Reinier Kramer
Affiliation: 
MPIM
Date: 
Mon, 2020-10-19 10:30 - 12:00

Following chapter 1 of Fulton's book, I will introduce cycles on schemes and define when cycles are rationally equivalent. I will show that rational equivalence behaves well with respect to certain maps between schemes and that these maps therefore descend to the cycle class groups or Chow groups.

https://bbb.mpim-bonn.mpg.de/b/rei-xh2-kg6
For password email to rkramer@mpim...

Divisors

Posted in
Speaker: 
Nitin Chidambaram
Affiliation: 
MPIM
Date: 
Mon, 2020-10-26 10:30 - 12:00

I will review the notion of Cartier divisors briefly and show that we can intersect Cartier divisors with arbitrary cycles in the Chow group. Then, I will show some properties satisfied by this intersection class, and finally discuss some applications (first Chern class and the Gysin map).

https://bbb.mpim-bonn.mpg.de/b/rei-xh2-kg6
For password email to rkramer@mpim...

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