Mixing
When is the blow up morphism flat?
$begingroup$
The question is as in the title: given a scheme $X$ and a closed subscheme $Z$ when is it true that the blow up morphism $Bl_ZX rightarrow X$ is flat?
I’m mainly concerned with $X$ being a smooth projective variety but a general answer would be appreciated.
My idea was to work affine locally and recover something from $Bl_0 mathbb{A}^n rightarrow mathbb{A}^n$ which it seems to me to be flat (but I might be wrong), but I wasn’t able to get far.
algebraic-geometry flatness blowup
asked Jan 12 at 19:35
FedericoFederico
888313
$endgroup$
add a comment |
$begingroup$
The question is as in the title: given a scheme $X$ and a closed subscheme $Z$ when is it true that the blow up morphism $Bl_ZX rightarrow X$ is flat?
I’m mainly concerned with $X$ being a smooth projective variety but a general answer would be appreciated.
My idea was to work affine locally and recover something from $Bl_0 mathbb{A}^n rightarrow mathbb{A}^n$ which it seems to me to be flat (but I might be wrong), but I wasn’t able to get far.
algebraic-geometry flatness blowup
asked Jan 12 at 19:35
FedericoFederico
888313
$endgroup$
add a comment |
$begingroup$
The question is as in the title: given a scheme $X$ and a closed subscheme $Z$ when is it true that the blow up morphism $Bl_ZX rightarrow X$ is flat?
I’m mainly concerned with $X$ being a smooth projective variety but a general answer would be appreciated.
My idea was to work affine locally and recover something from $Bl_0 mathbb{A}^n rightarrow mathbb{A}^n$ which it seems to me to be flat (but I might be wrong), but I wasn’t able to get far.
algebraic-geometry flatness blowup
asked Jan 12 at 19:35
FedericoFederico
888313
$endgroup$
The question is as in the title: given a scheme $X$ and a closed subscheme $Z$ when is it true that the blow up morphism $Bl_ZX rightarrow X$ is flat?
I’m mainly concerned with $X$ being a smooth projective variety but a general answer would be appreciated.
My idea was to work affine locally and recover something from $Bl_0 mathbb{A}^n rightarrow mathbb{A}^n$ which it seems to me to be flat (but I might be wrong), but I wasn’t able to get far.
algebraic-geometry flatness blowup
algebraic-geometry flatness blowup
asked Jan 12 at 19:35
FedericoFederico
888313
asked Jan 12 at 19:35
FedericoFederico
888313
asked Jan 12 at 19:35
FedericoFederico
888313
asked Jan 12 at 19:35
FedericoFederico
888313
asked Jan 12 at 19:35
FedericoFederico
888313
888313
add a comment |
add a comment |
1 Answer
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$begingroup$
If the blowup is non-trivial (when $Z$ has codimension $>1$ in $X$), then it is not flat. Indeed, away from the subvariety $Z$, the blowup is an isomorphism, so fibers over these points have Hilbert polynomial $P=1$. On the other hand, over a point in $Z$, the fiber is some projective space of positive dimension, and hence has a nontrivial Hilbert polynomial. Since a flat morphism's fibers have constant Hilbert polynomial (see Hartshorne, Chapter III, Proposition 9.9), we conclude that the blowup is not flat.
answered Jan 12 at 20:32
AndrewAndrew
3,1551617
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1 Answer
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1 Answer
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$begingroup$
If the blowup is non-trivial (when $Z$ has codimension $>1$ in $X$), then it is not flat. Indeed, away from the subvariety $Z$, the blowup is an isomorphism, so fibers over these points have Hilbert polynomial $P=1$. On the other hand, over a point in $Z$, the fiber is some projective space of positive dimension, and hence has a nontrivial Hilbert polynomial. Since a flat morphism's fibers have constant Hilbert polynomial (see Hartshorne, Chapter III, Proposition 9.9), we conclude that the blowup is not flat.
answered Jan 12 at 20:32
AndrewAndrew
3,1551617
$endgroup$
add a comment |
$begingroup$
If the blowup is non-trivial (when $Z$ has codimension $>1$ in $X$), then it is not flat. Indeed, away from the subvariety $Z$, the blowup is an isomorphism, so fibers over these points have Hilbert polynomial $P=1$. On the other hand, over a point in $Z$, the fiber is some projective space of positive dimension, and hence has a nontrivial Hilbert polynomial. Since a flat morphism's fibers have constant Hilbert polynomial (see Hartshorne, Chapter III, Proposition 9.9), we conclude that the blowup is not flat.
answered Jan 12 at 20:32
AndrewAndrew
3,1551617
$endgroup$
add a comment |
$begingroup$
If the blowup is non-trivial (when $Z$ has codimension $>1$ in $X$), then it is not flat. Indeed, away from the subvariety $Z$, the blowup is an isomorphism, so fibers over these points have Hilbert polynomial $P=1$. On the other hand, over a point in $Z$, the fiber is some projective space of positive dimension, and hence has a nontrivial Hilbert polynomial. Since a flat morphism's fibers have constant Hilbert polynomial (see Hartshorne, Chapter III, Proposition 9.9), we conclude that the blowup is not flat.
answered Jan 12 at 20:32
AndrewAndrew
3,1551617
$endgroup$
If the blowup is non-trivial (when $Z$ has codimension $>1$ in $X$), then it is not flat. Indeed, away from the subvariety $Z$, the blowup is an isomorphism, so fibers over these points have Hilbert polynomial $P=1$. On the other hand, over a point in $Z$, the fiber is some projective space of positive dimension, and hence has a nontrivial Hilbert polynomial. Since a flat morphism's fibers have constant Hilbert polynomial (see Hartshorne, Chapter III, Proposition 9.9), we conclude that the blowup is not flat.
answered Jan 12 at 20:32
AndrewAndrew
3,1551617
answered Jan 12 at 20:32
AndrewAndrew
3,1551617
answered Jan 12 at 20:32
AndrewAndrew
3,1551617
answered Jan 12 at 20:32
AndrewAndrew
3,1551617
3,1551617
add a comment |
add a comment |
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