Reading group “Intersection theory on stacks”

Between July 2020 and January 2021, I organised an online reading group on intersection theory on stacks, with a weekly session on Mondays from 10:30 to 12:00 (Bonn local time, GMT+1). The summer leg of this reading group was aimed at defining algebraic stacks. It covered preliminaries on category theory and algebraic geometry, Vistoli's Notes on Grothendieck topologies, fibered categories and descent theory, and material from Olsson's book Algebraic Spaces and Stacks and Canonaco's Lectures on algebraic stacks.
Half October, we started the autumn leg, focusing on intersection theory. The main focus was on intersection theory on schemes, mostly following Fulton's Intersection Theory. We also consider the theory on stacks, mostly following Vistoli's Intersection theory on algebraic stacks and on their moduli spaces. The final few lectures, in January, concerned the construction of virtual fundamental classes, using Battistella-Carocci-Manolache's Virtual Classes for the Working Mathematician as a guide.
Talks and recordings can be found here, but are only accessible with a password.

Talks and abstracts

Applications of virtual fundamental classes to enumerative geometry
January 25th, 2021
Speaker: Rosa Schwarz

The aim of this talk is to show why understanding the construction of a virtual fundamental class is useful: firstly we do this via computing these in some examples of moduli spaces of stable maps and secondly by linking these classes to Gromov-Witten invariants and thus to certain problems in enumerative geometry. I will start with a recap from the summer (17th August), when moduli spaces of stable maps were introduced, giving explicit examples along the way. Then I will follow the [BCM]-article that we have been studying to develop an obstruction theory and thus construct virtual classes for these spaces. I will end with a discussion of Gromov-Witten invariants.

Properties of obstruction theories and virtual classes
January 18th, 2021
Speaker: Reinier Kramer

After a recap of last week's constructions of perfect obstruction theories and virtual fundamental classes, I will discuss their properties with respect to deformation, and construct virtual pullbacks, which are bivariant classes. Then I will discuss the behaviour of virtual classes under pushforwards and give computational examples.

Defining perfect obstruction theories and virtual fundamental classes
January 11th, 2021
Speaker: Nitin Chidambaram

I will talk about obstruction bundles and use them to give a preliminary definition of virtual fundamental classes. We will use this notion to compute the virtual fundamental class explicitly in some examples. Then, we will define virtual fundamental classes more generally as in Behrend-Fantechi using the notion of perfect obstruction theories.

Bivariant intersection theory of stacks
December 21st, 2020
Speaker: Pim Spelier

Two weeks ago, we had a talk on Chow groups of stacks. Last week, we had a talk on bivariant intersection theory on schemes, i.e. operational Chow groups and Chow rings. This week, we will combine these two techniques to talk about bivariant intersection theory on stacks, where everything behaves much the same way, with a few subtleties. We will also compare the rational (bivariant) intersection theory on a stack with that of a moduli space. I will give many examples, to hopefully help develop some intuition about all these things.

Bivariant intersection theory
December 14th, 2020
Speaker: Campbell Wheeler

We will see how to generalise the notion of Gysin homomorphisms, which will define bivariant class groups. These will give us access to a unified framework to discuss both cohomology and homology. We will see that they come with many natural operations. We will also see natural spaces that Chern classes live in and discuss Poincaré duality as well as discussing how some of the theorems we previously proved generalise to bivariant class groups.

Defining intersection theory on stacks
December 7th, 2020
Speaker: Reinier Kramer

Using the machinery of both the summer and the fall parts of this reading group, I will define intersection theory on (Deligne-Mumford) stacks, using Vistoli's paper. I will commence by developing structural properties of stacks, such as the degree of a morphism. Furthermore, I will define a weak notion of moduli space for DM stacks, and go on to define intersection theory on stacks and prove naturality properties.

Excess and residual intersections
November 30th, 2020
Speaker: Rostislav Devyatov

In general, the intersection of subschemes need not be connected. I will give formulas for the equivalence of one connected component in the related intersection product. Furthermore, I will consider the case where the intersection of two subschemes contains a Cartier divisor of one of the subschemes, in which case it is also possible to define a complement, called the residual intersection class. I will give examples of applying both of these techniques.

Intersection multiplicities and Chow rings for non-singular varieties
November 23rd, 2020
Speaker: Rosa Schwarz

Last week, the intersection product was defined using the theory of cones. In the first past of this talk, we will show that under mild conditions, we can compute intersection multiplicities, certain coefficients in the intersection products, simply as the length of a module. We also give a clear criterion for when this intersection multiplicity is one. The second part of the talk, we show that in case of a non-singular variety, the constructed intersection product will yield a multiplication that makes the Chow group into a Chow ring. After this construction and some properties that hold in the smooth case, we compute this Chow ring in several examples. Finally, we can also rephrase the perhaps familiar Bézout's theorem in this context.

Definition and properties of intersection products
November 16th, 2020
Speaker: Pim Spelier

Using the machinery of cones, we will define the intersection product of a scheme and a variety. Passing through rational invariance, this defines so-called refined Gysin homomorphisms, acting on Chow groups, representing intersection with some fixed subscheme. These generalise constructions seen in previous talks, and we will see that they satisfy many nice properties.

Cones, Segre classes, and deformation to the normal cone
November 9th, 2020
Speaker: Alessandro Giacchetto

I will briefly introduce the notion of cones and define Segre classes in this context. I will then discuss the deformation to the normal cone of closed subschemes. The existence of such a deformation, together with the principle of continuity that intersection products should vary nicely in families, explains the prominent role to be played by the normal cone in constructing intersection products.

Vector bundles and Chern classes
November 2nd, 2020
Speaker: Campbell Wheeler

We will define higher Chern classes and Gysin homomorphisms. To do this we will first introduce Segre classes which turn out to be more natural for cones. These objects will all be important when we define higher intersections in the coming weeks.

October 26th, 2020
Speaker: Nitin Chidambaram

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).

Algebraic cycles and rational equivalence
October 19th, 2020
Speaker: Reinier Kramer

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.

Summer talks, on stacks

Moduli stacks of curves
August 17th, 2020
Speaker: Séverin Charbonnier

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.

Algebraic spaces and algebraic stacks
August 10th, 2020
Speaker: Praphulla Koushik

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”.

Definition and examples of stacks
August 3rd, 2020
Speaker: Campbell Wheeler

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.

Fibered categories (Part 2/2)
July 27th, 2020
Speaker: Alessandro Giacchetto

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.

Fibered categories (Part 1/2)
July 20th, 2020
Speaker: Praphulla Koushik

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.

Sites and sheaves
July 13th, 2020
Speaker: Reinier Kramer

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.

Recap of scheme theory
July 9th, 2020
Speaker: Reinier Kramer

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.

Category background for stacks
July 6th, 2020
Speaker: Reinier Kramer

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.