Let $X$ be a scheme corresponding to a smooth projective curve over $k=\bar{k}$.
Let $Div_X$ denote the set of effective Cartier divisors of $X$ and $Pic_X$ the set of line bundles over $X$. We can give scheme structures to these sets, as it is covered for example in this article. We can then look at the Abel-Jacobi as a morphism of schemes $$ A:Div_X \to Pic_X\;, \qquad D\mapsto\mathcal{O}_X(D) $$ My question is: is the Abel-JAcobi map a flat morphism?
I guess the smoothness of $X$ is necessary to hope for a $A$ to be flat... is my intuition correct?
By the way, do you think it is possible to answer this question (loosely) by a geometrical argument?