Can a smooth curve of any genus be realized on $\mathbb{CP}^1 \times \mathbb{CP}^1$?

Question

By the degree-genus formula $$g = \frac{(d-1)(d-2)}{2},$$ we know the possibility of the genus $g$ of a smooth curve on $\mathbb{CP}^2$ is limited by its degree $d$. What about a smooth curve on $\mathbb{CP}^1 \times \mathbb{CP}^1$? Can every genus $g\geq 0$ be realized?