If $X = C\times C' \subset \mathbb{P}^3$ for $C$ of genus $g$ and $C'$ of genus $g'$ (both smooth), then we know from Hartshorne exercise V.1.5 that $$ 8(g-1)(g'-1) = d(d-4)^2 $$ where $d = \text{deg}(X)$.
Is there a similar formula for $X = C\times C' \subset \mathbb{P}^N$ for $N$ arbitrary?