Anton at 18:58, 30 November 2011

2011-11-30T18:58:54Z

← Older revision		Revision as of 10:58, 30 November 2011
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	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 $X$.		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 $X$.


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Anton: Created page with "{{todo|add standard macros to mathjax config file}} Define essential dimension as follows. The essential dimension of an (algebraic) stack $\X$ is the smallest dimension of a sc..."

2011-11-15T19:30:17Z

Created page with "{{todo|add standard macros to mathjax config file}} Define essential dimension as follows. The essential dimension of an (algebraic) stack $\X$ is the smallest dimension of a sc..."

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{{todo|add standard macros to mathjax config file}}

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 $X$.

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Essential dimension - Revision history

Anton at 18:58, 30 November 2011

Anton: Created page with "{{todo|add standard macros to mathjax config file}} Define essential dimension as follows. The essential dimension of an (algebraic) stack $\X$ is the smallest dimension of a sc..."