Let $C$ be a smooth projective curve of genus $g > 0$ over an algebraically closed field. Suppose $r > 2g - 2$. Then given $r$ points $(P_1, \cdots, P_r) \in C^g$, Riemann-Roch implies that $l(P_1 + \cdots + P_r) = r + 1 - g > 0$.
Question Can $$L(P_1 + \cdots + P_r) \to (P_1, \cdots, P_r)$$ be made into a vector bundle over $C^r$? If not, are there special cases where it is indeed a vector bundle?
Thank you!