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.
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.
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.
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.
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.
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.
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".
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.
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.
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).