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Chandan Singh Dalawat
Chandan Singh Dalawat
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Of course not. The only smooth projective curve over $\mathbf{Z}$ is $\mathbf{P}_1$, and for every prime $p$ there is a smooth projective curve over $\mathbf{F}_p$ other than $\mathbf{P}_1$.

Addendum A bit of googling brings up the overview of Serre's work written on the occasion of his getting the Abel prize. It is mentioned there that recently (2005) Serre has improved his 1961 result mentioned in the comment below

by showing that if the variety can be lifted (as a flat scheme) to a local ring $A$, then $p.A=0$. The basic idea consists in transposing the problem to the context of finite groups.

Reference I've come across these notes by Yi Ouyang of a course by Luc Illusie on Topics in Algebraic Geometry ; the final section deals with Serre's example.

Of course not. The only smooth projective curve over $\mathbf{Z}$ is $\mathbf{P}_1$, and for every prime $p$ there is a smooth projective curve over $\mathbf{F}_p$ other than $\mathbf{P}_1$.

Addendum A bit of googling brings up the overview of Serre's work written on the occasion of his getting the Abel prize. It is mentioned there that recently (2005) Serre has improved his 1961 result mentioned in the comment below

by showing that if the variety can be lifted (as a flat scheme) to a local ring $A$, then $p.A=0$. The basic idea consists in transposing the problem to the context of finite groups.

Of course not. The only smooth projective curve over $\mathbf{Z}$ is $\mathbf{P}_1$, and for every prime $p$ there is a smooth projective curve over $\mathbf{F}_p$ other than $\mathbf{P}_1$.

Addendum A bit of googling brings up the overview of Serre's work written on the occasion of his getting the Abel prize. It is mentioned there that recently (2005) Serre has improved his 1961 result mentioned in the comment below

by showing that if the variety can be lifted (as a flat scheme) to a local ring $A$, then $p.A=0$. The basic idea consists in transposing the problem to the context of finite groups.

Reference I've come across these notes by Yi Ouyang of a course by Luc Illusie on Topics in Algebraic Geometry ; the final section deals with Serre's example.

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Chandan Singh Dalawat
Chandan Singh Dalawat
  • 18.7k
  • 4
  • 87
  • 153

Of course not. The only smooth projective curve over $\mathbf{Z}$ is $\mathbf{P}_1$, and for every prime $p$ there is a smooth projective curve over $\mathbf{F}_p$ other than $\mathbf{P}_1$.

Addendum A bit of googling brings up the overview of Serre's work written on the occasion of his getting the Abel prize. It is mentioned there that recently (2005) Serre has improved his 1961 result mentioned in the comment below

by showing that if the variety can be lifted (as a flat scheme) to a local ring $A$, then $p.A=0$. The basic idea consists in transposing the problem to the context of finite groups.

Of course not. The only smooth projective curve over $\mathbf{Z}$ is $\mathbf{P}_1$, and for every prime $p$ there is a smooth projective curve over $\mathbf{F}_p$ other than $\mathbf{P}_1$.

Of course not. The only smooth projective curve over $\mathbf{Z}$ is $\mathbf{P}_1$, and for every prime $p$ there is a smooth projective curve over $\mathbf{F}_p$ other than $\mathbf{P}_1$.

Addendum A bit of googling brings up the overview of Serre's work written on the occasion of his getting the Abel prize. It is mentioned there that recently (2005) Serre has improved his 1961 result mentioned in the comment below

by showing that if the variety can be lifted (as a flat scheme) to a local ring $A$, then $p.A=0$. The basic idea consists in transposing the problem to the context of finite groups.

Source Link
Chandan Singh Dalawat
Chandan Singh Dalawat
  • 18.7k
  • 4
  • 87
  • 153

Of course not. The only smooth projective curve over $\mathbf{Z}$ is $\mathbf{P}_1$, and for every prime $p$ there is a smooth projective curve over $\mathbf{F}_p$ other than $\mathbf{P}_1$.