Hi everyone.


Let $X$ be a stable curve over algebraically closed field $k$ with genus $g>1$, and $p$ is a nonsingular point of $X$. I want to prove the global section of dualizing sheaf is base point free, and I know this question is equivalent to 


$dimH^{0}(X,O\_{X}(p))=1$.


How to prove it?



a question about stable curve
If $X$ is a stable curve, and $X\_i$ a singular component, then a section $f \in H^0(X,\mathcal{O}\_X(p))$ gives a section in the blow-up (constant on the new $\mathbb{P}^1$), so we can assume that $X$ is semi-stable with non-singular components, and no $\mathbb{P}^1$ intersects in only one point.


We have two cases to consider:


1. If the component $X\_i$ of $p$ has positive genus, then a global section $f$ can be restricted to $X\_i$ where it gives the isomorphism to $\mathbb{P}^1$ as usual.
2. If $X\_i$ has genus zero, then it intersects the rest of the curve in at least two points $x\_1$ and $x\_2$. Since $f$ can be restricted to individual components, it must be constant on the rest of the curve, so we get $f(x\_1)=f(x\_2)$. Now $f-f(x\_1)$ has two zeroes, and thus $f$ has more than a single pole on $X\_i$, so we get a contradiction.


This also shows that the result does not hold on arbitrary semi-stable curves.


