Intersection Theory Class 6 Ravi

Intersection Theory Class 18

Intersection Theory Class 14

1. Where we are: Segre classes of vector bundles, and Segre classes of cones 1 1.1. Segre classes of cones 1 2. What the “functoriality of Segre classes of subschemes” buys us 2 2.1. The multiplicity of a variety along a subvariety 2 3. Deformation to the normal cone 3 3.1. The construction 3 4. Specialization to the normal cone 5 4.1. Gysin pullback for local complete intersections 6 4.2. Inte...

Intersection Theory Class 16 Ravi Vakil

We’ve covered a lot of ground so far. I want to remind you that we’ve essentially defined a very few things, and spent all our energy on showing that they behave well with respect to each other. In particular: proper pushforward, flat pullback, c·, s·, s·(X, Y). Gysin pullback for divisors; intersecting with pseudo-divisors. Gysin pullback for 0sections of vector bundles. We know how to calcula...

Intersection Theory Classes 20 and 21: Bivariant Intersection Theory

Intersection Theory Class 17

The cone of X in Y is in fact a vector bundle (as X ↪→ Y is a local complete intersection); call itNXY. The cone CWY toW in Y may be quite nasty; but we saw that CWY ↪→ g ∗NXY. Then we define X · V = s∗[CWV] where s : W → g∗NXY is the zero-section. (Recall that the Gysin pullback lets us map classes in a vector bundle to classes in the base, dropping the dimension by the rank. Algebraic black b...