A course on Intersection theory

Current status

• 2 hours lecture and 2 hours exercise per week.
• Timing: Monday, 08:00am-10:00am (SR 404) and Thursday, 04:00pm-06:00pm (Room 226)
• For any doubts or discussions, students can meet me any time in my office 436.

Brief description of the course

We then introduce the Chow groups of a variety and study the intersection product using the Chern classes of a vector bundle and deformation to the normal cone. The techniques and concepts used in this process have their own importance. Using intersection products, we prove that the direct sum of the Chow groups of a smooth variety is actually a ring, called the Chow ring (or intersection ring) of the variety. If time permits, we will also prove the Grothendieck-Riemann-Roch Theorem, which relates the Chow ring with K_0, the Grothendieck group of vector bundles on the smooth variety.

Structure of the course

Lectures:

• Lectures 1-3:

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  We introduced the concept of homogenization and dehomogenization of polynomials. We defined intersection multiplicity of two projective plane curves at a point and proved Bezout's theorem. We then defined the product of two plane projective curves, which do not have a common component, and proved Max Noether's fundamental theorem. We derived a number of applications. In particular, we proved the existence of an abelian group structure on a non-singular cubic.
• Lecture 4:

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  We recalled a number of basic definitions from algebraic geometry. For example, we revisited the definitions of locally ringed spaces, of algebraic schemes, of separated and proper morphisms, of closed and open subschemes and so on.
• Lectures 5-7:

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  We defined the Chow groups of an algebraic scheme. We proved the existence of proper push-forward and flat pull-back. We also proved that the pull-back map for an affine bundle is surjective. Using proper push-forward, we defined the degree of a zero-dimensional cycle class on a complete algebraic scheme.
• Lecture 8-9:

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  We defined Cartier divisors and pseudo-divisors on an algebraic scheme. Given a pseudo-divisor, we represented it by a Cartier divisor and then defined a cycle class of a pseudo-divisor. We also defined the pull-back of a pseudo-divisor. With the help of pseudo-divisors, we defined the intersection (as a cycle class) of a cycle with a Cartier divisor. We then proved a proposition about various properties of this intersection product. In particular, we proved a projection formula. This proposition is the heart of all results to follow.
• Lecture 10:

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  We proved the commutativity of intersection classes and as a corollary we obtained that intersection with a divisor kills the cycles which are rationally equivalent to the zero cycle. We then defined Chern class of a line bundle and Gysin map for divisors.
• Lecture 11:

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  We defined Segre classes and Chern classes of a vector bundle. Using splitting principle, we proved the Whitney summation formula. We then describe the Chow groups of a vector bundle and of a projective bundle in terms of the Chow groups of the base.
• Lecture 12:

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  Given a vector bundle, we defined the Gysin homomorphism as the inverse of the pull-back map and then gave a constructive formula for it. This formula is in terms of the top Chern class of the universal quotient bundle on the projective completion of the vector bundle. We then defined refined Gysin homomorphisms for regular closed embeddings and stated a number of propositions (without proof).
• Lecture 13:

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  We recalled what have been done untill this lecture. We defined the refined intersections for morphisms to smooth schemes and we stated their various properties (without proof). We then defined intersection multiplicities.
• Lecture 14:

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  For a smooth scheme, we defined intersection product and proved that the direct sum of the Chow groups, graded by codimension, defines a commutative graded ring, called as the Chow ring of the smooth scheme. Given a projective bundle on a smooth scheme, we proved the projective bundle formula. Using the formula for trivial projective bundle, we proved a generalized Bezout's theorem. We concluded the series of lectures by proving that the Bezout's theorem discussed in Lecture 2 is a corollary of the generalized Bezout's theorem.

Exercise Classes:

Exercise Class 1: We solved a number of exercises given in lecture 1.

Exercise Classes 2-4: We introduced the concept of modules of finite length and their length. We proved a number of lemmas about the same. Using them, we defined order functions. We also defined flat ring homomorphisms and how does length behave for flat local homomorphism of local Artinian rings.

Exercise Class 5: Besides solving the exercises given in Lecture 5, we saw the valuation criterion for properness.

Exercise Class 6: We proved a number of properties of normal domains and of flat morphisms.

Exercise Class 7: We proved that the Chow groups of an affine space are trivial (except the top one). We also proved that the i-Chow groups of a projective space is generated by the class of an i-dimensional linear subspace. We also proved a number of examples based on lectures 5-7.

Exercise Class 8: We defined vector bundles and locally free sheaves on an algebraic scheme and proved the correspondence between them. We defined Cartier divisors and proved their relation with line bundles (and equivalently with locally free sheaves of rank 1). Given a Cartier divisor, we associated a cycle with it and proved that the cycle associated with a principal Cartier divisor is rationally equivalent to the zero cycle.

Exercise Class 9: We studied the pull-back of a pseudo-divisor in more details. We then considered an example to compute the intersection of two divisors.

Exercise Class 10: We considered a number of applications of the results proved in Lecture 8-10. We then defined the cone and the projective cone associated with a locally free sheaf. We also defined the projective completion of a cone.

Exercise Class 11: Using deformation to the Normal cone, we defined the Gysin homomorphism for regular closed embeddings.

Exercise Class 12: Besides sovling exercises given in Lecture 12, we proved that various definitions of Gysin homomophisms agree.

Exercise Class 13: We defined the sheaf of differentials and smooth morpshims. We then proved that a section of a smooth morphism is always a regular embedding and in particular, the graph (morphism) of a morphism to a smooth scheme is a regular embedding.

Prerequisite

References

• W. Fulton: Algebraic curves, An introduction to algebraic geometry
• W. Fulton: Intersection theory (second edition)
• D. Eisenbud and J. Harris: 3264 and all that