How is the natural evaluation map $V\otimes O_C \longrightarrow \pi_{2,*}(\pi_1^*L\otimes O_{C\times C}/J_\Delta^{r+1})$, where $V \subset H^0(C,L), L$ a line bundle on $C$, defined?
I am a little confused about $V\otimes O_C$; what is it tensoring over?
Thank you for any hints.
By pushforward-pullback adjunction it is enough to construct a map $$ \pi_2^*(V \otimes O_C) = V \otimes O_{C\times C} \to \pi_1^*L \otimes (O_{C \times C}/J^{r+1}) $$ (it is wrong to write $\Delta^*$ there). This map, in its turn, is a composition of the map $$ V \otimes O_{C\times C} = \pi_1^*(V \otimes O_C) \to \pi_1^*L $$ and of the map $O_{C \otimes C} \to O_{C \times C}/J^{r+1}$ tensored with $\pi_1^*L$.