Questions tagged [frobenius-splitting]

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Blow-ups of $ F $-regular varieties at points in general position and finite generation of the Cox ring

A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}_{X} $-linear map $ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{...
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Is the Cox ring of a $ \mathbb{Q} $-factorial, $ F $-regular, Mori dream space $ F $-regular?

A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}_{X} $-linear map $ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{...
  • 483
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Example of projective, $ F $-regular variety $ X $ and smooth sub-variety $ Y $ such that $ \operatorname{Bl}_{Y}(X) $ is not $ F $-regular?

A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}_{X} $-linear map $ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{...
  • 483
1 vote
1 answer
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Does there exist a point $ x $ of an affine toric variety $ U_{\sigma} $ such that $ x $ is not compatibly split?

A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}_{X} $-linear map $ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{...
  • 483
2 votes
0 answers
76 views

When is a smooth point of a projective, simplicial, toric variety $ X_{\Sigma} $ compatibly $ F $-split?

A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}_{X} $-linear map $ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{...
  • 483
2 votes
0 answers
115 views

Does anyone know an example of a non-singular, globally $ F $-regular variety $ X $ for which generic smoothness does not hold?

Let us denote the Frobenius endomorphism of a variety $ X $ by $ F $. A variety $ X $ over a field $ k $ of positive characteristic is globally $ F $-regular if for every effective Weil divisor $ D $,...
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