Suppose that the self product $E \times E$ of an elliptic curve $E$ contains a compact Riemann surface $C$ of genus $2$.
For points $a, b \in E$, we have to $D = a \times E+E \times b$ is ample divisor in $E\times E$.
My question is: Does anyone see any problem if there is $p$ in $E\times E$ such that $C= D + p$ ?
What is the absurdity if this happens?
Thank you!