In this 3rd semester of algebraic geometry, we will study vector bundles and algebraic cycles on algebraic varieties. We will first recall the main facts on divisors, linear equivalence, line bundles and the isomorphism of the Picard group with the group of divisors modulo linear equivalence. We will then look at the representability of the Picard group via the Picard variety.
The next step is to introduce cycles of higher codimension, the relation of rational equivalence and the Grothendieck group of vector bundles. Assuming a good intersection theory for cycles modulo rational equivalence on a smooth variety, we will recall Grothendieck's construction of Chern classes of vector bundles, give the modern proof of the Grothendieck-Riemann-Roch and go over a number of applications.
We will then fill in the missing step: the existence of a good intersection product, using Fulton's method of intersection with a divisor followed by the deformation to the normal bundle. If time permits, we will look at a number results on zero-cycles, such as Mumford's proof of the infinite dimensionality for zero-cycles on a surface with p_g>0, or Roitmann's theorem on the torsion zero-cycles. Other topics that we may cover include some aspects of enumerative geometry or representabiilty results.
Instructors: Marc Levine and Daniel Harrer
The first meeting is on 17.10.2016. Interested students who have a schedule conflict should send an email to Marc Levine (email@example.com).