Define what is meant by a rational map from a projective variety $V \subset \mathbb{P}^{n}$ to $\mathbb{P}^{m}$. What is a regular point of a rational map?

Consider the rational map $\phi: \mathbb{P}^{2}-\rightarrow \mathbb{P}^{2}$ given by

$$\left(X_{0}: X_{1}: X_{2}\right) \mapsto\left(X_{1} X_{2}: X_{0} X_{2}: X_{0} X_{1}\right)$$

Show that $\phi$ is not regular at the points $(1: 0: 0),(0: 1: 0),(0: 0: 1)$ and that it is regular elsewhere, and that it is a birational map from $\mathbb{P}^{2}$ to itself.

Let $V \subset \mathbb{P}^{2}$ be the plane curve given by the vanishing of the polynomial $X_{0}^{2} X_{1}^{3}+X_{1}^{2} X_{2}^{3}+X_{2}^{2} X_{0}^{3}$ over a field of characteristic zero. Show that $V$ is irreducible, and that $\phi$ determines a birational equivalence between $V$ and a nonsingular plane quartic.