Hi everyone.

Let $X$ be a stable curve over algebraically closed field $k$ with genus $g>1$, and $p$ is any 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?

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?

Hi everyone.

Let $X$ be a stable curve over algebraically closed field $k$, and $p$ is any point of $X$,

my question is:

$dimH^{0}(X,O_{X}(p))=1$ ?

Thank you very much.

Hi everyone.

Let $X$ be a stable curve over algebraically closed field $k$ with genus $g>1$, and $p$ is any 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?