Skip to main content
MathOverflow

Return to Question

added 336 characters in body
Source Link
user111251
user111251

Is there a skyscraper group scheme?

Let S$S$ 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?

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 $G$?

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?

Source Link
user111251
user111251

"skyscraper group scheme"

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 $G$?