TITLE: Indecomposable vector bundles on elliptic curves
QUESTION [5 upvotes]: Let $X$ be a smooth complex projective curve of genus 3,
$E$ an elliptic curve, and $f: X \to E$ a finite map of 
degree 2. Let $L$ be a line bundle on $X$, and $R^0f_*(L)$
its direct image. Question: is $R^0f_*(L)$ indecomposable?

REPLY [6 votes]: This depends on the choice of $L$. For instance, if $L = \mathcal{O}_X$ then $f_*L \cong \mathcal{O}_E \oplus L_E$, where $\deg(L_E) = -2$. 
On the other hand, if $L$ is a general line bundle of degree 1, then $f_*L$ is indecomposable.
If $L = \mathcal{O}(P)$ for some $P \in X$ the pushforward $f_*L$ is decomposable. Indeed, on the one hand, $f_*L$ is a vector bundle of rank 2 and degree $-1$ (this follows from Riemann-Roch), on the other hand, it has a global section. Therefore, it comes in an exact sequence
$$
0 \to \mathcal{O}_E \to f_*L \to M \to 0,
$$
where $M$ is a line bundle of degree $-1$. But
$$
Ext^1(M,\mathcal{O}_E) = H^1(E,M^\vee) = 0
$$
because the degree of $M^\vee$ is 1, hence the sequence splits and 
$$
f_*L \cong \mathcal{O}_E \oplus M.
$$