Unirational

A variety $X$ is unirational if there is a map $U \rightarrow X$ 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 $g \leq 14$. Moreover, since unirational implies Kodaira dimension $-\infty$, the result of Eisenbud, Harris, and Mumford shows that $\mathcal{M}_g$ is not unirational for $g \geq 24$.