What are some explicit examples (e.g., by explicitly describing its Hopf algebra) of finite unramified group schemes? (Ie, the sort of group schemes which appear as automorphism groups of objects parametrized by Deligne-mumford stacks).

For example, I'd be interested in finite unramified group schemes over $\mathbb{C}[[t]]$ which are not flat - e.g., they are generically etale, but the cardinality of its special fiber may jump.

Is there a good reference for such objects? Are there structure theorems for such things? (For example, does every finite unramified group scheme over $\mathbb{C}[[t]]$ admit a "maximal finite etale quotient"?)

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    $\begingroup$ For a non-flat example, simply take the ring $\mathbb C[[t]][x]/(x^n-1,t(x-1))$, whose spectrum is a closed subgroup of $\mu_n$. Its generic fibre is trivial, but the special fibre is $\mu_n$. $\endgroup$ Jun 11, 2017 at 6:23
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    $\begingroup$ For a geometric example similar to Remy's example: Fix $d\geq 3$ and $n\geq 1$. Let $F:\mathcal X\to H_{n,d}$ be the universal family of smooth hypersurfaces of degree $d$ in $\mathbb P^{n+1}$ over the (non-projective) Hilbert scheme $H_{n,d}$. The "general" fibre of $F$ is a smooth projective hypersurface with no non-trivial (linear) automorphisms.... $\endgroup$ Jun 12, 2017 at 12:27
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    $\begingroup$ ...However, there are certainly smooth hypersurfaces with non-trivial automorphisms (e.g., the Fermat hypersurface). Thus, let $f:X\to \mathbb C[[t]]$ be a smooth hypersurface of degree $d$ in $\mathbb P^{n+1}$ such that over "t=0" you have a Fermat hypersurface, whereas the generic fibre has no non-trivial automorphisms. Then, the group scheme $Lin(f)\to Spec \mathbb {C}[[t]]$ is finite unramified, but not flat. (Maybe it would be easier to just consider a family of smooth proper curves of genus 2, or a family of elliptic curves over $\mathbb{C}[[t]]$ whose generic fibre has $j \neq 0,1728$.) $\endgroup$ Jun 12, 2017 at 12:27


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