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Schemes in which every affine open subset is principal

Schemes covered by finitely-many affine open subsetsdoes every affine closed subscheme have an open affine neighborhood?Is every integral point of an arithmetic scheme contained in an affine open set?Principal open sets of affine schemesLine bundles on affine schemesBirational affine schemesOpen sets in the spectrum of a Dedekind domain are affineAffine Schemes and Basic Open SetsCan scheme be covered by affine schemes like varieties?Strength of “every affine scheme is compact”

1

$begingroup$

What are some known high dimensional projective schemes in which every affine open subset is principal?

If $X$ is a Dedekind scheme with torsion class group, then the answer is true.

algebraic-geometry commutative-algebra

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asked yesterday

zzyzzy

2,6331420

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What is your definition of principal for projective schemes?
  $endgroup$
  – Mohan
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Mohan in $Proj(S)$ we can call $D_+(f)$ principal open subsets.
  $endgroup$
  – zzy
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If $X$ is smooth (say over an algebraically closed field) and Picard group has rank one, what you say will be true. If Picard group has rank greater than one, it will not be true.
  $endgroup$
  – Mohan
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Mohan Thank you, how to see that?
  $endgroup$
  – zzy
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  By the way, I should have asked for one more clarification. While $S$ determines $X$, the converse is not true. Are you fixing the $S$ or allowing it to be any $S$ such that the proj is $X$?
  $endgroup$
  – Mohan
  yesterday
  

  
  
  
  

 | 
show 3 more comments

1

$begingroup$

What are some known high dimensional projective schemes in which every affine open subset is principal?

If $X$ is a Dedekind scheme with torsion class group, then the answer is true.

algebraic-geometry commutative-algebra

share|cite|improve this question

asked yesterday

zzyzzy

2,6331420

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What is your definition of principal for projective schemes?
  $endgroup$
  – Mohan
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Mohan in $Proj(S)$ we can call $D_+(f)$ principal open subsets.
  $endgroup$
  – zzy
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If $X$ is smooth (say over an algebraically closed field) and Picard group has rank one, what you say will be true. If Picard group has rank greater than one, it will not be true.
  $endgroup$
  – Mohan
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Mohan Thank you, how to see that?
  $endgroup$
  – zzy
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  By the way, I should have asked for one more clarification. While $S$ determines $X$, the converse is not true. Are you fixing the $S$ or allowing it to be any $S$ such that the proj is $X$?
  $endgroup$
  – Mohan
  yesterday
  

  
  
  
  

 | 
show 3 more comments

$begingroup$

What are some known high dimensional projective schemes in which every affine open subset is principal?

If $X$ is a Dedekind scheme with torsion class group, then the answer is true.

algebraic-geometry commutative-algebra

share|cite|improve this question

asked yesterday

zzyzzy

2,6331420

$endgroup$

share|cite|improve this question

asked yesterday

zzyzzy

2,6331420

share|cite|improve this question

asked yesterday

zzyzzy

2,6331420

asked yesterday

zzyzzy

2,6331420

asked yesterday

zzyzzy

2,6331420