Suppose $X$ is a smooth projective variety over $\mathbb C$. Then under what conditions, the natural map $H^0(X,mK_X) \times H^0(X,nK_X) \to H^0(X,(m+n)K_X)$ 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.

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.

Suppose $X$ is a smooth projective variety over $\mathbb C$. Then under what conditions, the natural map $H^0(X,mK_X) \times H^0(X,nK_X) \to H^0(X,(m+n)K_X)$ 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 $\oplus_{i=1}^m H^0(X, K_X) \to H^0(X, mK_X)$ to be surjective when $m \geq 3$.

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

Suppose $X$ is a smooth projective variety over $\mathbb C$. Then under what conditions, the natural map $H^0(X,mK_X) \times H^0(X,nK_X) \to H^0(X,(m+n)K_X)$ 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.