Pushforwad bundle alond a degree 2 map from $P^1$ to $P^1$

Question

Suppose $f:P^1\rightarrow P^1$ is a degree 2 morphism. Let $L$ be a line bundle on $P^1$ (which is equivalent to a $O(n)$), then what is $f_*L$? As for an open set $U$ we can see the preimage is a disjoint of two pieces of open sets, if $U$ is small enough, so the image is a rank 2 bundle. I guess that it should be $O(n)\oplus O(n)$, as we can calculate the dimension of sections.

Answer 1

First, $$ f_*O \cong O \oplus O(-1). $$ Indeed, this should be a rank 2 vector bundle with $H^0 = \Bbbk$ and $H^1 = 0$, by Grothendieck theorem any vector bundle on $\mathbb{P}^1$ is a sum of line bundles, and so the only option we have is the one written in the right hand side above. Similarly, $$ f_*O(-1) \cong O(-1) \oplus O(-1). $$ Finally, from the above and projection formula it follows that $$ f_*O(2n) \cong O(n) \oplus O(n-1), \qquad f_*O(2n-1) \cong O(n-1) \oplus O(n-1). $$