Anders Buch
Enumerative Geometry
Enumerative geometry concerns determining the number of geometric objects of a specified type that satisfy a list of conditions. Classical examples include the two lines meeting four general lines in projective 3-space, and the 27 lines contained in a smooth cubic surface. The standard strategy for solving enumerative problems is to start with a moduli space parameterizing the type of objects to be counted, and then hope that each condition corresponds to a subvariety of the moduli space defined by polynomial equations. In this case the problem is reduced to finding the number of points in the intersection of a list of subvarieties, which often amounts to a calculation in the cohomology ring of the moduli space. We will start with an introduction to intersection theory and Chow cohomology, and then examine the subproblems one encounters when employing the standard strategy. This includes understanding the cohomology ring of the moduli space, as well as the classes in this ring defined by the subvarieties corresponding to the conditions. The last question will lead us to degeneracy locus formulas such as the classical Thom-Porteous formula and generalizations.
Fulton, Intersection Theory (for the start of the course).
Some knowledge of algebraic geometry is an advantage, but students willing to take some technical facts for granted will be able to follow the course if they are familiar with any type of geometry, including manifolds.