Patrick Lei

Intersection Theory (Spring 2021)

References are

Jan 22
Caleb Ji
Rational equivalence.
I will introduce the notions of cycles and rational equivalence, which generalize the notions of divisors and linear equivalence. They give rise to Chow groups, which play to role of homology groups on schemes. The main theorem of this talk is that rational equivalence of cycles is preserved under the pushforward of a proper morphism. Bezout’s theorem is a consequence of this theorem. I will end by discussing flat pullback of cycles.
Reference: [F], Chapter 1
Jan 29
Avi Zeff
Intersecting with divisors and the first chern class
We define Weil and Cartier divisors and pseudo-divisors, and show how intersecting with these divisors yields maps \(A_k(X) \to A_{k-1}(X)\). We show that as operators on \(A_*(X)\) this action of divisors by intersection is commutative.
Feb 05
Alex Xu
Chern Classes and Segre Classes of Vector Bundles
In this talk we will discuss the construction and functorial properties of Chern and Segre classes in intersection theory following chapter 3 of Fulton. An emphasis will be placed on using the functorial properties for computation and we will work through several examples. If time permits, we will attempt to work through some of the high octane examples that lead to a geometric interpretation for these algebraic gadgets.
Feb 12
Patrick Lei
Cones: because not every coherent sheaf is locally free
We will discuss what a cone is, then define the Segre class of a cone, then define Segre classes of subvarieties and consider their properties, and then discuss deformation to the normal cone. Classical examples will be used to illustrate the theory.
Reference: [F], Chapters 4,5
Feb 19
Caleb Ji
Chern classes and intersection products
In this talk, we will revisit the notion of Chern classes in algebraic geometry and apply them to enumerative problems. then we will review the moving lemma and sketch the construction of the intersection product in the Chow ring.
Feb 26
Problem Session
Mar 05
No seminar (spring break)
Mar 12
Nicolás Vilches
Families of algebraic cycles
We will discuss families of algebraic cycles: the specialization of a class on the total space of a family. The relation between the original class and its specializations will be discussed extensively, such as the conservation of number. After this, we will show how to apply this machinery to classical problems in enumerative geometry.
Reference: our friend [F], Chapter 10
Mar 19
Patrick Lei
Doing Italian-style algebraic geometry rigorously
We will define intersection multiplicities and then define the Chow ring. For smooth varieties, the Chow ring behaves formally like cohomology in some ways. Finally, we will discuss Bézout’s theorem, which has a very short proof in our language and then discusss some classical examples.
Reference: [F], Chapters 7,8
Mar 26
Morena Porzio
The Grothendieck-Riemann-Roch theorem
We begin by introducing the terminology behind the statement and then state the GRR theorem for proper morphisms of smooth varieties. We will see why it is a generalization of RR for curves and HRR for surfaces. Then we focus on the proof of the result in the important special case \(\mathbb{P}^n \to pt\).
Apr 02
Patrick Lei
Fine moduli memes for 1-categorical teens
We will discuss the moduli space of stable curves of genus 0 with \(n\) marked points and its intersection theory, following Keel. We will give a nice presentation of its Chow ring in terms of boundary divisors.
Reference: Keel, Intersection theory of moduli space of stable N-pointed curves of genus zero
Apr 09
Patrick Lei
Do we even need derived categories?
We will state Serre’s intersection formula which computes intersection multiplicities using the derived tensor product. Then we will give some cases where we do not need derived categories to compute intersection multiplicities.
Apr 16
Caleb Ji
Motivic cohomology
Following Voevodsky’s lectures, we introduce the category of correspondences and the notion of presheaves of transfers, which allows us to define motivic cohomology. Among other things, these groups specialize to the higher Chow groups defined by Bloch. We give a broad overview of the relations between motivic cohomology and algebraic K-theory, motives, and arithmetic geometry.