Consider a moduli problem $\mathcal{M}$. Assume, at each point $x$, the associated deformation problem $\mathcal{M}_x$ has a tangent-obstruction theory. It follows that $\mathcal{M}_x$ has a hull $spec\, R_x \to \mathcal{M}_x$. I think it means, for every point $x$, there is a chart $spec \, R_x$ etale over the point $x$. But in general, $\mathcal{M}$ is not Deligne-Mumford, and it does not admit an etale chart. I am confused about this. Does that mean the miniversality is not an open condition? I guess the picture is like this. Assume $x$ is one point with positive dimensional automorphism. Then the dimension of the tangent space $T_x$ is bigger, so if I extend the corresponding $R_x$ to an etale neighbourhood, it might be versal in the neighbourhood, but not miniversal. Am I correct?
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$\begingroup$ Usually the phrase "is etale over the point $x$" means more than "is formally etale precisely at the point $x$". It means "there is an open neighborhood on which the morphism is etale". To pass from "formally etale at one point $x$" to "etale on an open neighborhood", Artin used the hypothesis that the functor / stack is limit preserving / finitely presented in an essential way. You can probably make counterexamples by considering functors / stacks that are not limit preserving. $\endgroup$– Jason StarrCommented Mar 12, 2015 at 6:18
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$\begingroup$ I See. Thanks for the comment. I will need to study Artin's paper more closely to understand the obstruction to being etale on an open neighborhood. $\endgroup$– user38276Commented Mar 13, 2015 at 16:46
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