Suppose \(Y\) is a smooth variety over which lies a vector bundle \(E\) of rank \(r\) given by a locally free sheaf \(\mathscr E\). Recall that sections \(s\) of \(E\) over \(Y\) are in bijection with global sections of \(\mathscr E^\vee\) (Hartshorne II.5.18), and let \(s\) be such a section. Let \(\iota:Z\hookrightarrow Y\) be (the closed embedding of) the zero-set \(Z\) of \(s\), and let \(\mathcal I\) be the corresponding sheaf of ideals in \(\mathcal O_Y\).
Definition
The normal cone of \(Z\) in \(Y\) is
\begin{equation} C_{Z}Y = \underline{\Spec}\left(\bigoplus_{k=0}^{\infty}\mathcal I^k/\mathcal I^{k+1}\right). \end{equation}This works for any closed embedding \(Z\hookrightarrow Y\). However, since \(Z\) is the zero set of \(s \in \mathscr E^\vee\)…