Given a line bundle over a scheme, we can form the first Chern class of the line bundle. We’ll use this to sort of duck tape a Segre class together out of a vector bundle over \(X\).
Definition
Let \(X\) be a scheme and \(E\) a vector bundle on \(X\) of rank \(e+1\). Let \(P= \Proj_XE\) be the projectivization of \(E\) – so that \(P\) locally looks like \(X\times \mathbb P^e\) – and \(\mathcal O_E(1)\) the tautological line bundle over \(P\). Let \(p:P\to X\) be the projection down to \(X\). For any class \(\alpha \in A_k(X)\), define \(s_i(E)\cap \alpha\) to be
\begin{align*} s_i(E)\cap \alpha = p_*\big(c_1(\mathcal O_E(1))^{e+i} \cap p^*(\alpha)\big) \end{align*}where
- \(p^*\) is the flat pullback to \(P\)
- \(p_*\) is the proper pushforward back to \(X\), and
- \(c_1(\mathcal O_E(1))^{e+i}\) is the iterated first Chern class of \(\mathcal O_E(1)\) on \(P\).
Basically, pull \(\alpha\) back to \(P\), picking up \(\rank E - 1 = e\) (fiber dimension of \(p\)) dimension so that \(p^*\alpha \in A_{k+e}(P)\); then intersect with the \(e+i\)th iterate of the first Chern class of \(P\) to cut the dimension down by \(e+i\) and land in \(A_{k - i}(P)\) and finally push back down to \(X\), an operation which preserves the degree of the cycle.