Suppose $X$ is a smooth projective variety over $\mathbb C$. Then under what conditions, the natural map $H^0(X,mK\_X) \otimes H^0(X,nK\_X) \to H^0(X,(m+n)K\_X)$ for $m, n \in \mathbb{Z}\_{>0}$ is surjective?


My case is particularly simple: $X$ is a smooth **curve** of genus greater than $1$, and I wish $\otimes\_{i=1}^m H^0(X, K\_X) \to H^0(X, mK\_X)$ to be surjective when $m \geq 3$. 


However, I was unable to show this or give a conterexample.



When is the map $H^0(X,mK_X) \times H^0(X,nK_X) \to H^0(X,(m+n)K_X)$ surjective?
I assume you mean $H^0(X, K\_X)^{\otimes m}$ rather than $\oplus\_{i=1}^m H^0(X, K\_X)$. If $X$ is a smooth projective connected complex curve of genus $g \geq 2$, then the map
$$H^0(X, K\_X)^{\otimes m} \longrightarrow H^0(X, m K\_X),$$
is surjective for any $m \geq 0$, as long as $X$ is not hyperelliptic. This is a theorem of M. Noether.


