Show that \(\PP^n\cross \PP^m\) is rational.
Solution
On the dense open set \(x_0y_0\neq 0\), define the rational map \(\PP^n\cross\PP^m\dashrightarrow\PP^{n+m}\) by \[ [x_0: x_1: \ldots: x_n] \cross [y_0: y_1: \ldots: y_m] \mapsto \left[1: \frac{x_1}{x_0}: \frac{x_2}{x_0}: \ldots: \frac{x_n}{x_0}: \frac{y_1}{y_0}: \frac{y_2}{y_0}: \ldots: \frac{y_m}{y_0}\right] .\] It has inverse, defined on \(z_0\neq 0\), \[ [1: z_1: \ldots: z_{n+m}] \mapsto [1: z_1: \ldots: z_n] \cross [1: z_{n+1}: \ldots: z_{n+m}] ,\] so \(\PP^n\cross\PP^m\) is birational to \(\PP^{n+m}\).