Unirational variety

Remarks

Equivalently, the function field $k(X)$ embeds over $k$ into a purely transcendental extension $k(t_1, \ldots, t_N)$. Example: every rational variety is unirational, since a birational map is dominant. Example: the curve $y^2 = x^3$ in $\mathbb{A}^2$ is unirational. The map $t \mapsto (t^2, t^3)$ is a dominant morphism from $\mathbb{A}^1$, an open subset of $\mathbb{P}^1$, since $(t^3)^2 = (t^2)^3$ and its image is the whole curve. Non-example: a smooth projective curve $E$ of genus $1$ over $\C$ is not unirational. A dominant rational map $\mathbb{P}^N \dashrightarrow E$ restricts to a nonconstant rational map on a general line, and this extends to a nonconstant morphism $f \colon \mathbb{P}^1 \to E$. The Riemann–Hurwitz formula would then give $-2 = 2g(\mathbb{P}^1) - 2 \ge \deg f \cdot (2g(E) - 2) = 0$, which is false.