Let $S$ be a locally noetherian scheme, $Y$ a locally noetherien $S$-scheme and $X$ an abelian scheme over $S$. It is known that the map between groups $Hom(Y,X) \to Hom(Pic(X/S),Pic(Y/S)), f \mapsto f^*$ is quadratic, i.e. we have
$(f+g+h)^* - (f + g)^* - (f + h)^* - (h + h)^* + f^* + g^* + h^* = 0$.
However, $f \mapsto f^\*$ is not linear. Is there an easy example for $(f+g)^* \neq f^* + g^*$?
This is related to the order of the functor $X \mapsto Pic(X/S)$. The above inequality would include a nontrivial line bundle on $X \times_S X$, which is trivial on the both closed subschemes $X \times 0, 0 \times X$. I'm also interested in an easy example for this phenomenon.