Chern Classes And Segre Classes Topic Page

To do intersection theory with a line bundle \(L\) over a scheme \(X\), it’d make sense to associate some closed set \(D\subset X\) to \(L\) and then intersect stuff with \(D\). Intuitively, this \(D\) should be codimension \(1\), and if \(V\subset X\) is some subvariety, then we’d hope the closed set \(D\cap V\) corresponds to the line bundle \(L|_V\) in \(V\). In a perfect world, we’d have

\begin{align*} \{\text{Line bundles}\} \leftrightarrow \{\text{Cartier divisors}\} \leftrightarrow \{\text{Weil divisors}\}, \end{align*}

in which case this process would be exactly the first Chern class of \(L\); that is, \(c_1(L)\) is simply a map which takes a \(k\)-cycle \(\alpha\in A_k(X)\) and intersects it with its associated Weil divisor to attain a \(A_{k-1}(X)\) cycle. However, we don’t live in a perfect world, as instead we have

\begin{align*} \{\text{Line bundles}\} \hookleftarrow\{\text{Cartier divisors}\} \hookrightarrow \{\text{Weil divisors}\}. \end{align*}

This association follows from the following nodes.

• Cartier divisors
• Each Cartier divisor uniquely determines a locally principal Weil divisor
• Cartier divisors uniquely determine line bundles

To do intersection theory with line bundles, we need a way to associate…

• Pseudo divisor

We can then define the first Chern class of a line bundle and we retrieve desirable properties.

• The Chern class of a line bundle

At this point in the development of intersection theory we have proper pushforward, flat pullback and the first Chern class of a line bundle. We use all of these to define the Segre class of a vector bundle.

• Segre class of vector bundle

Then we prove some properties of the Segre class.

• Properties of Segre classes

Note that this only serves to define the Segre class of a vector bundle; we must treat the case of cones slightly differently.