Unirational

A variety $X$ is unirational if there is a map $U\to X$ from an open subset $U\subset {\mathbb{A}}^N$ of some affine space whose image contains a dense, open subset of $X$. For instance, ${\mathcal{M}}_g$ is unirational means that there is a family of curves on an open subset of affine space which contains a general curve of genus $g$. This is known to be the case for $g\le 14$. Moreover, since unirational implies Kodaira dimension $-\infty$, the result of Eisenbud, Harris, and Mumford shows that ${\mathcal{M}}_g$ is not unirational for $g\ge 24$.