Let $k$ be an algebraically closed field, $C$ a genus $g$ smooth projective curve over $k$ and $A=Pic^d(C)\subseteq Jac(C)$ the moduli variety of all deg $d$ line bundles. For any degree $d$ line bundle $L$ on $C$ (i.e $L \in A(k)$), we can consider $f_i(L)=\text{dim}_kH^i(C,L)$ for $i=0,1$. By Riemann-Roch theorem, we know $f_0(L)-f_1(L)=d+1-g$. My question is, what do we know about $f_0(L)$ as a function on $A(k)$? For general $L$, what is $f_0(L)$? Is $f_0$ constant on some open subset or upper semi-continuous?
Example: If $g=1$ and $d=0$, then $f_0(L)=0$ for non-trivial $L$. If not, there is a nonzero section on $L$. Note $C$ is a elliptic curve in this case, and multiplication by $[-1]$ gives $[-1]^*L=L^{-1}$ hence both $L$ and $L^{−1}$ have a non-trivial section so $L$ is trivial.
What about the general case?
Edit: Thanks for the answer, could we even compute $f_0(L)$ like the $d=0$ case?
For any $g$ one has $$ H^0(C,L) = \begin{cases} k, & \text{if $L \cong \mathcal{O}_C$},\\ 0, & \text{otherwise}. \end{cases} $$ So, $f_0$ for $ = 0$ is zero away from the origin of $Pic^0(C)$.
For arbitrary $d$ the function $f_0$ is upper semi-continuous. The corresponding stratification of $Pic^d(C)$ is not so easy to describe. The subvariety, where $f_0 > 0$ is the image of the natural map $$ \gamma \colon S^d(C) \to Pic^d(C). $$ The subvariety $f_0 > 1$ is the critical locus for this map, and more generally, the subvariety $f_0 \ge i$ coincides with the locus of points, over which the dimensions of the fibers of $\gamma$ is at least $i$.