# Is this incidence variety in $\mathbb{P}^2 \times \mathbb{P}^2$ isomorphic to a variety in $\mathbb{P}^1 \times \mathbb{P}^1$?

I have an incidence variety $X = \{(p,\ell) \in C \times D^* : p \in \ell\} \subset \mathbb{P}^2 \times \mathbb{P}^2$, where $C = Z(f) \subset \mathbb{P}^2$ and $D^* = Z(g^*) \subset \mathbb{P}^2$ are two conics ($D^*$ is the dual conic of the conic $D = Z(g)$).

From Hartshorne exercise I.3.1(c) I know that every conic is isomorphic to $\mathbb{P}^1$. Can I use this to write $X$ as a variety in $\mathbb{P}^1 \times \mathbb{P}^1$? If so, what equations would define this variety?