Essential dimension

Define essential dimension as follows. The essential dimension of an (algebraic) stack $\X$ is the smallest dimension of a scheme $X$ with the following property: there is a morphism $X\to \X$ so that for any field $k$, any map $Spec(k)\to \X$, factors through $X$. Note that this agrees with the usual notion of essential dimension (?) by considering the generic point of $X$. In particular, if $Y\to \X$ is any morphism from a scheme, we have that the map from the generic point of $Y$ factors through $X$, so (with appropriate finite presentation hypotheses), there is a dense open of $Y$ where the map to $\X$ factors through $X$.