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Is there a skyscraper group scheme?

Let $S$ be a DVR. Is there a group scheme $\mathcal{G}$ over $S$ which is generically {1} trivial i.e, identity group, but at the closed point some nontrivial group $G$?

For example: Let $C$ be a curve over $S$ whose generic fibre is smooth of genus $g$ with no automorphism but the closed fibre is a semistable curve(which has nontrivial automorphisms). Now consider the group of $S$-automorphisms of this curve $C$ over $S$? My question: Does this curve C over S has any non trivial S-automorphism?

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2 Answers 2

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Yes.

Let $\mathbb{A}^{1,2}$ be the affine line with a double origin. Consider the natural map $\mathbb{A}^{1,2}\to \mathbb{A}^1$. (To define this map, let $0_1$ and $0_2$ be the origins in $\mathbb{A}^{1,2}$. The above map sends any $x\neq 0_1, 0_2$ to $x$. It sends $0_1$ and $0_2$ to the origin in $\mathbb{A}^1$.)

The above morphism realizes $\mathbb{A}^{1,2}$ as a quasi-finite flat group scheme over $\mathbb{A}^1$. (It is not a separated group scheme over $\mathbb{A}^1$, of course.)

Note that the generic fibre of this group scheme is the trivial group. The fibre over the origin is the group $\mathbb{Z}/2\mathbb{Z}$.

To get to the situation you desire, let $\mathrm{Spec} \mathbb{C}[[t]]\to \mathbb{A}^{1}$ be a dominant map whose image contains the origin. Now base-change the group scheme $\mathbb{A}^{1,2}\to \mathbb{A}^1$ along this morphism to get a group scheme $$ G\to \mathrm{Spec} \mathbb{C}[[t]]$$ with trivial generic fibre and a non-trivial special fibre. (Here $G=\mathbb{A}^{1,2}\times_{\mathbb{A}^1} \mathrm{Spec} \mathbb{C}[[t]]$.)

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  • $\begingroup$ @ZSun This answer only answers your first question. As Laurent Moret-Bailly points out in his answer to both your questions, the above non-separated group scheme is never the automorphism group scheme of a stable curve $X$ over $S$. Even worse, if $(X,L)\to S$ is a polarized flat proper finitely presented morphism, then $Aut(X,L)\to S$ is an affine (hence separated) group scheme. Now, some varieties $X$ come with a "canonical'' (or "anti-canonical'') polarization $L$ (e.g., for stable curves of genus at least two you can take $L=\omega_X$) in which case $Aut(X) = Aut(X,L)$ is affine. $\endgroup$
    – Ariyan Javanpeykar
    Commented Jan 17, 2018 at 19:17
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Yes: the following examplpe is quite different from the example by Ariyan, although the generic and closed fibers are the same as his.

Start with the constant group scheme $A:=(\mathbb{Z}/2\mathbb{Z})_S$, and consider the closed subscheme which is the union of the zero section and the closed point of the other section. This is immediately seen to be a subgroup scheme of $A$, with generic fiber $0$ and closed fiber $\mathbb{Z}/2\mathbb{Z}$. It is finite over $S$ (in particular separated) but of course not flat.

You can construct infinitely many variants by replacing the closed point by any infinitesimal neighborhood of it.

If you take a stable curve $C$ over $S$, it it known that its automorphis functor $\mathscr{G}:={\underline{\mathrm{Aut}}}(C/S)$ is a finite unramified $S$-group scheme, hence it "looks like" the above example, rather than Ariyan's nonseparated one. In particular ($K$ being the fraction field), the restriction homomorphism $\mathscr{G}(S)\to \mathscr{G}(K)$ is bijective, so if $C_K$ has no $K$-automorphisms then if $C$ has no $S$-automorphisms.

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    $\begingroup$ This is a very nice answer. To "realize" your group scheme in "nature", let $X\to S$ be a smooth proper genus three curve whose generic fibre $X_{\eta}$ is a "generic" smooth quartic curve in $\mathbb{P}^2$ (with no non-trivial automorphisms), and whose special fibre is a "generic" hyperelliptic curve of genus three (with automorphism group $\mathbb{Z}/2\mathbb{Z}$). $\endgroup$
    – Ariyan Javanpeykar
    Commented Jan 17, 2018 at 20:17

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