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missing (implicit) hypotheses
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anonymous
anonymous
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It is a basic fact that $H^n(X, F) = 0$ if $X$ is noetherian affine, $n > 0$, and $F$ a quasi-coherent sheaf.

If $Y \to X$ is a blow-up of a smooth variety in a smooth center, then then exceptional divisor is a projective bundle over the center, and so $H^n(Y, \mathcal{O}_Y) = H^n(X, \mathcal{O}_X)$. (right?)

I have not seen any examples of blow-ups of smooth varieties (with arbitrary center) whose fibres are not unions of projective spaces (sub-question: do such blow-ups exist?).

So I am lead to wonder: is it always true that $H^n(Y, \mathcal{O}_Y) = 0$ if $f: Y \to X$ is birational, $Y$ is integral, and $X$ is smooth and affine?

Edit: I mean't to include the hypothesis that $f$ is proper, and that $n$ is any integer $> 0$.

It is a basic fact that $H^n(X, F) = 0$ if $X$ is noetherian affine, $n > 0$, and $F$ a quasi-coherent sheaf.

If $Y \to X$ is a blow-up of a smooth variety in a smooth center, then then exceptional divisor is a projective bundle over the center, and so $H^n(Y, \mathcal{O}_Y) = H^n(X, \mathcal{O}_X)$. (right?)

I have not seen any examples of blow-ups of smooth varieties (with arbitrary center) whose fibres are not unions of projective spaces (sub-question: do such blow-ups exist?).

So I am lead to wonder: is it always true that $H^n(Y, \mathcal{O}_Y) = 0$ if $f: Y \to X$ is birational, $Y$ is integral, and $X$ is smooth and affine?

It is a basic fact that $H^n(X, F) = 0$ if $X$ is noetherian affine, $n > 0$, and $F$ a quasi-coherent sheaf.

If $Y \to X$ is a blow-up of a smooth variety in a smooth center, then then exceptional divisor is a projective bundle over the center, and so $H^n(Y, \mathcal{O}_Y) = H^n(X, \mathcal{O}_X)$. (right?)

I have not seen any examples of blow-ups of smooth varieties (with arbitrary center) whose fibres are not unions of projective spaces (sub-question: do such blow-ups exist?).

So I am lead to wonder: is it always true that $H^n(Y, \mathcal{O}_Y) = 0$ if $f: Y \to X$ is birational, $Y$ is integral, and $X$ is smooth and affine?

Edit: I mean't to include the hypothesis that $f$ is proper, and that $n$ is any integer $> 0$.

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anonymous
anonymous
  • 73
  • 3

Hn(X, OX) = 0 for X birational to a regular affine variety?

It is a basic fact that $H^n(X, F) = 0$ if $X$ is noetherian affine, $n > 0$, and $F$ a quasi-coherent sheaf.

If $Y \to X$ is a blow-up of a smooth variety in a smooth center, then then exceptional divisor is a projective bundle over the center, and so $H^n(Y, \mathcal{O}_Y) = H^n(X, \mathcal{O}_X)$. (right?)

I have not seen any examples of blow-ups of smooth varieties (with arbitrary center) whose fibres are not unions of projective spaces (sub-question: do such blow-ups exist?).

So I am lead to wonder: is it always true that $H^n(Y, \mathcal{O}_Y) = 0$ if $f: Y \to X$ is birational, $Y$ is integral, and $X$ is smooth and affine?