Segre classes and Chern classes ( Overview )

For a vector bundle over a scheme , consider the projective bundle . Then we can use the canonical line bundle on to define the $i-th$ Segre class of as follows : for a cycle , we pull it back, and we let the first Chern class of the canonical line bundle act on the pullback cycle times ( so that we get the dimensions as expected), and then pushforward the result. Here we utilize the property that the projection from the projective bundle is both flat and proper.
The Segre classes have the following properties inherited from that of Chern class of line bundle :
- Projection formula
- Flat pull back
- Commutativity of classes
It also has vanishing property : that is, the Segre classes whose orders are negative or bigger than the dimension of are all trivial

Now we can define formally the Segre power series, and then take the inverse to get the Chern polynomials, which then gives us the Chern classes. The Chern classes also have the properties like projection formula, flat pulback, commutativity, vanishing, and one other important property is the Whitney sum formula, which states that the Chern polynomial of the “sum” of two vector bundles ( the middle term is the short exact sequences ) is the product of those of other two.

Pushforward and flat pull-back of intersections

These are two quite useful properties of intersecting with divisors.
1- Projection formula :
Let be proper. Then if is a divisor on , we want to pull back , intersect it with a cycle in , and then push it back. The projection formula gives a shortcut to that : we can just pushforward the cycle, and then intersect it with .
.

( Where is the induced map from the relevant subschemes : from to . This is important, because we have to adhere to where we define our intersection classes!)

2- Flat pull back
Let be flat. Then if $latex$ F is a divisor on , we want to pull back the intersection of a cycle with . So this formula tells us that we can just pull back and and do the intersection on . Again, we need to be a bit careful to restrict maps to relevant subscheme.

The idea of proof :
1- We can assume is a surjective proper morphism of varieties, and the cycle is just itself.

2-

The nice thing about projection formula and flat pull back is that it allows us to define Chern classes of vector bundle : we pushforward the Chern classes (intersection classes) defined by the canonical line bundle on the corresponding projective bundle. The projection map from the projective bundle is both proper and flat.

Intersecting with pseudo-divisors

Suppose we have a pseudo-divisor on a scheme . Then for each subvariety of , we can define the intersection by pulling back ( restricting ) the corresponding line bundle with section of on to , which will in turn determine a divisor supported in . In case the support of contains , then we can only define that intersection divisor up to rational equivalence ( at level of homology class ). Otherwise, it is uniquely determined at the level of cycle. Note that we want to define a class in , and it is different from defining a class in . A cycle can in fact represent the 0 class in the latter but not in the former. Plus, later we shall need commutativity of intersection, thus we want to define a class in so that no matter in what order we intersect, the classes always be defined in the same space.

Now we can extend the definition by linearity to all the cycles. There is a special case where we can definite intersections at cycle level : when the line bundle of is trivial in a neighborhood of . In that case, in the only situation where the previous failed to determine a cycle i.e. when the support of a subvariety is contained , we can just define the intersection to be the 0 cycle.

As we should expect, if the line bundle that comes with is trivial, then intersecting with always yield 0 on the groups . In fact, if is any variety, then the restriction of on is trivial, hence it defines the 0 homology class. Later, we will know that intersecting with divisors yield a homomorphism of Chow groups ( i.e., intersecting with 0 homology class yield a homology class, as a corollary of commutativity of intersection ).

There are two properties of intersection with divisors which are useful but somewhat not quite trivial to prove :
1- Projection formula
2- Flat pullback

In some situation, when we are sure that Cartier divisors can be pulled backed ( i.e., the image scheme is not contained in the support of the divisor, or when the map is flat ( pull back of nonzero divisors are nonzero divisors , hence we can pull back the ration functions that define the Cartier divisor under flat map ). In general, since the support of a Cartier divisor is contained in that of the pseudo-divisor it represents, the equality involving two intersection classes (with Cartier divisor ) is better, since the classes are defined in smaller spaces.