Let $X$ be a complex torus with a fixed positive definite line bundle $L$. I'd like to show that the generic element of $|L|$ is reduced.
I tried to use the fact that given $x_1 , \dots x_n$ points on $X$ such that $x_1 + \dots x_n=0$ we have $$t^*_{x_1}D \otimes \dots \otimes t^*_{x_n}D \cong D^n .$$
Also, I tried to use the fact that if $L$ is of type $(d_1, \dots d_g)$ then there exists $M \in Pic(X)$ such that $M^{d_1}=L$ , but I did not manage to get anything.
In particular, I did not get how to properly use the fact that $L$ is positive definite (neither to understand whi this hypothesis is necessary).