Assume we are given two smooth curves $C_1$ and $C_2$ over an algebraically closed field $k$. It is known that divisorial correspondences between them correspond to homomorphisms between their Jacobians. In particular there exist pairs of curves without nontrivial correspondences between them. Still my construction appears to yield such a correspondence for two arbitrary curves. So I am wondering if my construction is wrong or if all my correspondences are trivial. Could you please help me? I proceed as follows:

First I embed the function fields $K_1,K_2$ of the curves into a finite galois subextension of $k(t)^{alg}/k(t)$. This extension corresponds to a curve $C$ with nonconstant morphisms $f:C\rightarrow C_1$ and $g:C\rightarrow C_2$. The Galois group G of $C/C_1$ operates on C making $f$ into the quotient morphism. By composition with $g$ the operation induces several morphisms $g_i:C\rightarrow C_2$.

Now, I consider the sum of the graphs of the $g_i$ in $C\times C_2$. This is a G-stable Divisor descending to $C_1\times C_2$. Correcting with vertical divisors yields the nontrivial divisorial correspondence.