All the notes linked here are preparatory in nature, and we as the organizer did not proofread them at all. As such, there may be mistakes that the speakers did not realize and/or did not bother to correct. Readers should read at their own risk.
[Ful84] W. Fulton, Intersection Theory, Ergebnisse der Mathematik, 3.folge. Band 2, SpringerVerlag, Berlin Heidelberg 1984.
Date: May 2, 2025
Place: MATH 4-130
Speaker: Prabhat Devkota
Title: Grothendieck-Riemann-Roch Theorem, III
Abstract: Abstract: The Grothendieck-Riemann-Roch theorem for smooth varieties states that the homomorphism \tau(--) = ch(--) td(X) from the Grothendieck ring to the rational Chow ring of X "commutes" with the proper pushforward. However, for a singular variety, the appropriate analog of this homomorphism is not quite clear. In this session, we will see that we can use the localized Chern character introduced last talk to cook up such a homomorphism that allows us to formulate and prove the analog of Grothendieck-Riemann-Roch for singular (quasi-projective) varieties. Afterwards, we will discuss the specialization/application of the theorem to common cases like birational morphism/resolution of singularities, flat families, etc. If time remains, we will also get a glimpse into the proof in the general (possibly not quasi-projective) case.
Date: April 25, 2025
Speaker: Connor Lehmacher
Title: Grothendieck-Riemann-Roch Theorem, II
Abstract: We will continue our discussion of Grothendieck Riemann Roch, extending the result to (possibly singular) schemes. First we give an application of smooth GRR to count the number of lines on a cubic surface. After some more discussion of bivariant intersection theory, we state GRR for an arbitrary algebraic scheme. This leads to the striking corollary that the rational Grothendieck group (of sheaves) is isomorphic to the rational Chow group. Time allowing we will discuss MacPherson’s graph construction which is needed to prove GRR.
Date: April 18, 2025
Speaker: Prabhat Devkota
Title: Grothendieck-Riemann-Roch Theorem, I
Abstract: The Grothendieck-Riemann-Roch Theorem is a fundamental result in algebraic geometry which relates the Chern character of a coherent sheaf with that of its pushforward under a proper morphism. We will prove the theorem for a proper morphism of non-singular quasiprojective varieties. When the morphism is an inclusion, we can restate the theorem using the total Chern class, yielding a formula with integer coefficients (aptly called the Riemann-Roch without denominators). This formula will then be used to compute the Chern classes of the tangent bundle of a blow-up along a non-singular subvariety.
Date: April 11, 2025
Speaker: Ze Yun
Title: Correspondences and Bivariant Intersection Theory
Abstract: I'll first talk about correspondences, which are like Fourier-Mukai transforms on Chow groups. After giving its functorial properties, I'll present an example. Then I'll give some definitions in bivariant intersection theory, which will be used in Grothendieck-Riemann-Roch for singular varieties.
Date: April 4, 2025
Speaker: Connor Lehmacher
Title: Schubert Calculus
Abstract: Today we will discuss the intersection theory for Grassmanians. First we will review the Plücker embedding. Then we will describe Schubert cycles which are the generators of the Chow ring of a Grassmanian. We will give concrete calculations for Gr(2,4), and then discuss the general intersection theory; specifically Pieri’s and Giambelli’s formulas.
Date: March 28, 2025
Speaker: Dingchang Zhou
Title: Excess Intersection and Residual Intersection
Abstract: Today we will review the principles in constructing the intersection product, and use the intersection product to compute the excess intersection and residual intersection formulas. We will also go through examples to demonstrate the formulas.
Reference: Chapter 9 of [Ful84]
Date: March 14, 2025
Speaker: Mattew Huynh
Title: The intersection Ring and Bezout's Theorem
Abstract: We will define a ring structure on the Chow group of a nonsingular variety where the product is given by intersection of cycles. Next we will discuss the problem of assigning multiplicities of intersections and how this is dealt with using our formalism. Finally, we will discuss Bezout's Theorem as well as a classical application of it.
Reference: Chapter 7,8 of [Ful84]
Computation of Two planes in P^4
Date: March 7, 2025
Speaker: Nate Tausik
Title: The Intersection Product, I
Abstract: In this talk, we will first describe the deformation to the normal cone and the specialization homomorphism. This construction is the last piece of background we need to define the intersection product V∙X of a k-dimensional scheme V mapping to a scheme Y and a regularly imbedded codimension d subscheme X of Y. After defining the product, along with the related notions of distinguished components and Gysin homomorphisms, we will establish some basic properties of the intersection product including proper pushforward, flat pullback, excess intersection, commutativity and functoriality. We will then see how the product can be generalized from regular imbeddings to LCI morphisms, and finally discuss the special case where V is the blowup of Y along X. Time permitting, we will establish some properties of intersection multiplicities, the coefficients of the distinguished components in the intersection class V∙X.
Reference: Chapter 5,6,7 of [Ful84]
Date: February 28, 2025
Speaker: Connor Lehmacher
Title: Segre Classes of Cones
Abstract: In this talk, we will generalize the Segre class from vector bundles to cones. First I will review cones, discussing how they generalize line bundles and blow-ups. After making the definition for the Segre class, we specialize to the case of normal cones. As an application we will see how to define algebraic multiplicity of a subvariety and time allowing I will discuss the case of normal cones for base loci of a linear system.
Reference: Chapter 4 of [Ful84]
Date: February 21, 2025
Speaker: Maximilian Hofmann
Title: Chern classes on vector bundles
Abstract: In this talk, we will define Chern classes as the inverse to the so-called total Segre class, which can be constructed from just the first Chern class already introduced during last week's talk. We will then collect the most important properties of Chern classes with a special focus on the splitting principle. Using these tools, we will be able to compute Chern classes in several interesting examples. Finally, we will investigate the relationship between the Chow ring of a vector bundle and the Chow ring of the base space, leading to a more general definition of the Gysin homomorphism.
Reference: Chapter 3 of [Ful84]
Date: February 14, 2025
Speaker: Shuo Gao
Title: Divisors
Abstract: In this talk, we discuss the notions of Weil, Cartier and pseudo- divisors, and their relations. Motivated by topology, we find that support of the divisor plays an important role when we are intersecting the divisor with cycles. As such, such intersection maps are well defined on the divisor level, but not always so on the linear equivalence classes. Another interesting feature of intersection theory is that the Chow groups of a subvariety could very well be more complicated and interesting than the Chow groups of the ambient space. To access this feature, we will define two maps to be generalized in later talks: intersection with the (first) Chern class(es) and the Gysin maps.
Reference: Chapter 2 of [Ful84]
Date: February 7, 2025
Speaker: Shuo Gao
Title: Rational Equivalence and Chow Groups
Abstract: We start with the definitions and first properties of rational equivalences. We will then define the first Chern class of divisors in the Chow ring. If time permits, we will also talk about the Chern class of general vector bundles. An overview of intersection theory towards our goal - Grothendieck-Riemann-Roch - will be provided in the end.
Reference: Chapter 1 of [Ful84]