Let $\pi: X \to Y$ be a map between reasonable schemes. Let $L$ be a locally free sheaf on $X$, i.e. a line bundle. It's not always the case that $\pi_* L$ is a vector bundle but sometimes it is. For example, if $\text{dim}_{k(y)} H^0 (X_y, L_y)$ is a constant function of $y \in Y$ then $\pi_* L$ is a vector bundle of rank $\text{dim}_{k(y)} H^0 (X_y, L_y).$
My question is the following: what is the geometric intuition behind this vector bundle? Are there any examples where I can explicitly see it geometrically? For example, if $X \to Y$ is a covering map of degree $n$ (that is, etale of degree $n$), then $X_y$ is just a disjoint union of $n$ points, therefore $\text{dim}_{k(y)} H^0 (X_y, L_y)$ is indeed constant. How to understand this vector bundle geometrically in this example?