Showing that image of a rational map is dense

Question

I have the rational map $$\varphi: \mathbb{P}^2 \dashrightarrow \mathbb{P}^3, [X_0, X_1, X_2] \mapsto [X_0^2, X_0 X_1, X_0 X_2, X_1 X_2]$$ This map is defined on $U_0 \cup (U_1 \cap U_2)$ which is open and dense, now I would like to show that its image is dense in $Q = V(X_0 X_3 - X_1 X_2) $. How can I do that?

Answer 1

Look at the restriction of $\varphi$ to $D(X_0)\subset\Bbb P^2$. This lands in $D(X_0)\cap Q\subset Q$, and this morphism of affine varieties is of the form $\Bbb A^2\to V(X_3-X_1X_2)\subset\Bbb A^3$ by $(X_1,X_2)\mapsto (X_1,X_2,X_1X_2)$. This is an isomorphism with inverse $V(X_3-X_1X_2)\to \Bbb A^2$ by $(X_1,X_2,X_3)\mapsto (X_1,X_2)$. Since $Q$ is irreducible, any nonempty open subset is dense, and we're done.