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.



Picard functor is not linear
Let $E$ be an elliptic curve; take $f = g = \mathrm{id}\_E$. Then $f+g$ is multiplication by $2$, and has degree $4$; hence if $L$ is an invertible sheaf on $E$, the sheaf $(f+g)^\*L$ has degree $4 \deg L$, while $f^\*L \otimes g^\*L = L^{\otimes 2}$ has degree $2\deg L$.


