Say we have a reduced, irreducible, complete curve $X$ over a field $k$ and a sheaf $\mathcal F$ on $X$. I am interested in understanding the cohomology groups of $\mathcal F$.
By flat base change, we can assume $k$ is algebraically closed and by Grothendieck's vanishing theorem, $H^i(X,\mathcal F)$ is zero for $i\ge 2$. So the interesting information (if any) is reflected in $H^0$ and $H^1$, in particular $\chi(\mathcal F)=h^0-h^1$. $H^0$ is just the global sections.
If anything up to here is incorrect please correct me. I am not fully comfortable with the topic.
This leaves $H^1$. How should I think of $H^1$ (or $h^1$)? I know it is the ''obstruction'' to exactness of $\Gamma(X,\cdot)$, but that doesn't quite help me visualize it.
In the particular instance of what I am reading, $\mathcal F$ is a torsion-free coherent $\mathcal O_X$-Module and it is assumed that $h^0=h^1=0$. If the above is correct, all cohomology groups thus vanish (since we are on a curve). Does this ''mean'' anything? (or should I just take it as a given and continue reading until hopefully it becomes clear)
Thank you.