Pullback Degree Formula

Lemma

1. Let \(f:C\to C'\) be a morphism of smooth curves and let \(\mathcal L\) be a line bundle on \(C'\). Then

\begin{align*} \deg(f^*\mathcal L) = \deg(f) \cdot \deg (\mathcal L). \end{align*}

1. Let \(:C\to \mathbb P^n\) be a closed embedding of a smooth curve \(C\) into \(\mathbb P^n\). Then for any line bundle \(\mathcal L\) on \(\mathbb P^n\),

\begin{align*} \deg (\mathcal L|_C) = \deg (C) \cdot \deg (\mathcal L). \end{align*}