The pullback line bundle restricted on the exceptional divisor is trivial

Question

Let $\sigma:\hat X\to X$ be the blow up of a point $x\in X$, denote the exceptional divisors $\sigma^{-1}(x)$ by $E$. $L\to X$ is a line bundle. Then we have a pullback line bundle $\sigma^*L\to\hat X$. Then why is the restriction of $\sigma^*L|_E\to E$ trivial?

Answer 1

You're pulling back $L$ along $\sigma$ and then pulling back the result along the inclusion $i:E\to\hat X$. That's the same as pulling $L$ back along the composite map $\sigma\circ i$, and that composite map is the constant map $E\to X$ sending all of $E$ to the point $x$. And pulling any bundle back along a constant map always gives you a trivial bundle.