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$.

Jeffrey Herschel Giansiracusa 2005-06-27