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user6319
user6319
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It is well known that Kodaira gave an essentially topological classification of the possible singular fibers of elliptic fibrations according to their type:

https://en.wikipedia.org/wiki/Elliptic_surface#Kodaira.27s_table_of_singular_fibers

Les us assume that we have two elliptic fibrations $\pi\colon X\to S$, $\pi'\colon X'\to S'$ and consider two singular fibers $X_s\hookrightarrow X$, $X'_{s'}\hookrightarrow X'$ over closed points such that the Kodaira type of $X_s$ is the same as the Kodaira type of $X'_{s'}$.

What I'm asking is whether it is always true or not that $X_s$ and $X'_{s'}$ are isomorphic as schemes with their naturally induced closed subscheme structure.

EDIT: By an elliptic fibration $\pi\colon X\to S$, I understand:

$X$ is a smooth algebraic surface. $S$ is a complete smooth algebraic curve over an algebraically closed field, $\eta$ is its generic point. $\pi$ is a projective morphism, and the generic fiber $X_\eta$ is a geometrically integral smooth algebraic curve of arithmetic genus $1$.

It is well known that Kodaira gave an essentially topological classification of the possible singular fibers of elliptic fibrations according to their type:

https://en.wikipedia.org/wiki/Elliptic_surface#Kodaira.27s_table_of_singular_fibers

Les us assume that we have two elliptic fibrations $\pi\colon X\to S$, $\pi'\colon X'\to S'$ and consider two singular fibers $X_s\hookrightarrow X$, $X'_{s'}\hookrightarrow X'$ over closed points such that the Kodaira type of $X_s$ is the same as the Kodaira type of $X'_{s'}$.

What I'm asking is whether it is always true or not that $X_s$ and $X'_{s'}$ are isomorphic as schemes with their naturally induced closed subscheme structure.

It is well known that Kodaira gave an essentially topological classification of the possible singular fibers of elliptic fibrations according to their type:

https://en.wikipedia.org/wiki/Elliptic_surface#Kodaira.27s_table_of_singular_fibers

Les us assume that we have two elliptic fibrations $\pi\colon X\to S$, $\pi'\colon X'\to S'$ and consider two singular fibers $X_s\hookrightarrow X$, $X'_{s'}\hookrightarrow X'$ over closed points such that the Kodaira type of $X_s$ is the same as the Kodaira type of $X'_{s'}$.

What I'm asking is whether it is always true or not that $X_s$ and $X'_{s'}$ are isomorphic as schemes with their naturally induced closed subscheme structure.

EDIT: By an elliptic fibration $\pi\colon X\to S$, I understand:

$X$ is a smooth algebraic surface. $S$ is a complete smooth algebraic curve over an algebraically closed field, $\eta$ is its generic point. $\pi$ is a projective morphism, and the generic fiber $X_\eta$ is a geometrically integral smooth algebraic curve of arithmetic genus $1$.

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user6319
user6319
  • 179
  • 3

Is it true that specialsingular fibers of elliptic fibrations that have the same Kodaira type are isomorphic schemes?

It is well known that Kodaira gave an essentially topological classification of the possible specialsingular fibers of elliptic fibrations according to their type:

https://en.wikipedia.org/wiki/Elliptic_surface#Kodaira.27s_table_of_singular_fibers

Les us assume that we have two elliptic fibrations $\pi\colon X\to S$, $\pi'\colon X'\to S'$ and consider two specialsingular fibers $X_s\hookrightarrow X$, $X'_{s'}\hookrightarrow X'$ over closed points such that the Kodaira type of $X_s$ is the same as the Kodaira type of $X'_{s'}$.

What I'm asking is whether it is always true or not that $X_s$ and $X'_{s'}$ are isomorphic as schemes with their naturally induced closed subscheme structure.

Is it true that special fibers of elliptic fibrations that have the same Kodaira type are isomorphic schemes?

It is well known that Kodaira gave an essentially topological classification of the possible special fibers of elliptic fibrations according to their type:

https://en.wikipedia.org/wiki/Elliptic_surface#Kodaira.27s_table_of_singular_fibers

Les us assume that we have two elliptic fibrations $\pi\colon X\to S$, $\pi'\colon X'\to S'$ and consider two special fibers $X_s\hookrightarrow X$, $X'_{s'}\hookrightarrow X'$ over closed points such that the Kodaira type of $X_s$ is the same as the Kodaira type of $X'_{s'}$.

What I'm asking is whether it is true or not that $X_s$ and $X'_{s'}$ are isomorphic as schemes with their naturally induced closed subscheme structure.

Is it true that singular fibers of elliptic fibrations that have the same Kodaira type are isomorphic schemes?

It is well known that Kodaira gave an essentially topological classification of the possible singular fibers of elliptic fibrations according to their type:

https://en.wikipedia.org/wiki/Elliptic_surface#Kodaira.27s_table_of_singular_fibers

Les us assume that we have two elliptic fibrations $\pi\colon X\to S$, $\pi'\colon X'\to S'$ and consider two singular fibers $X_s\hookrightarrow X$, $X'_{s'}\hookrightarrow X'$ over closed points such that the Kodaira type of $X_s$ is the same as the Kodaira type of $X'_{s'}$.

What I'm asking is whether it is always true or not that $X_s$ and $X'_{s'}$ are isomorphic as schemes with their naturally induced closed subscheme structure.

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user6319
user6319
  • 179
  • 3

Is it true that special fibers of elliptic fibrations that have the same Kodaira type are isomorphic schemes?

It is well known that Kodaira gave an essentially topological classification of the possible special fibers of elliptic fibrations according to their type:

https://en.wikipedia.org/wiki/Elliptic_surface#Kodaira.27s_table_of_singular_fibers

Les us assume that we have two elliptic fibrations $\pi\colon X\to S$, $\pi'\colon X'\to S'$ and consider two special fibers $X_s\hookrightarrow X$, $X'_{s'}\hookrightarrow X'$ over closed points such that the Kodaira type of $X_s$ is the same as the Kodaira type of $X'_{s'}$.

What I'm asking is whether it is true or not that $X_s$ and $X'_{s'}$ are isomorphic as schemes with their naturally induced closed subscheme structure.