Cohomology of tangent sheaf of a hypersurfaceDimension of irreducible components of tangent bundle Big...
Cohomology of tangent sheaf of a hypersurface
Dimension of irreducible components of tangent bundle Big tangent bundleEmbedding a projective curve in a smooth surface (using a Bertini theorem)Cremona transformationsSpecial linear sections of a hypersurfaceVanishing of sheaf cohomology with compact supportCohomology of tangent sheaf of a singular hypersurfaceHessian of an hypersurfaceIntersection numbers in $mathbb{P}^1$-bundlesHow to write down the connection morphism in the long exact sequence in Čech cohomology explicitly in this specific case?
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Let $Xsubsetmathbb{P}^n$ be an irreducible and reduced hypersurface of degree $d$. How can one explicitly compute the dimension of the vector spaces $H^0(X,T_X),H^1(X,T_X),H^2(X,T_X)$? Here $T_X$ is the tangent sheaf of $X$.
For instance $h^0(X,T_X)$ gives the dimension of the automorphism group of $X$.
ag.algebraic-geometry sheaf-theory projective-geometry birational-geometry sheaf-cohomology
asked yesterday
user125056user125056
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add a comment |
$begingroup$
Let $Xsubsetmathbb{P}^n$ be an irreducible and reduced hypersurface of degree $d$. How can one explicitly compute the dimension of the vector spaces $H^0(X,T_X),H^1(X,T_X),H^2(X,T_X)$? Here $T_X$ is the tangent sheaf of $X$.
For instance $h^0(X,T_X)$ gives the dimension of the automorphism group of $X$.
ag.algebraic-geometry sheaf-theory projective-geometry birational-geometry sheaf-cohomology
asked yesterday
user125056user125056
261
New contributor
user125056 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Let $Xsubsetmathbb{P}^n$ be an irreducible and reduced hypersurface of degree $d$. How can one explicitly compute the dimension of the vector spaces $H^0(X,T_X),H^1(X,T_X),H^2(X,T_X)$? Here $T_X$ is the tangent sheaf of $X$.
For instance $h^0(X,T_X)$ gives the dimension of the automorphism group of $X$.
ag.algebraic-geometry sheaf-theory projective-geometry birational-geometry sheaf-cohomology
asked yesterday
user125056user125056
261
New contributor
user125056 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Let $Xsubsetmathbb{P}^n$ be an irreducible and reduced hypersurface of degree $d$. How can one explicitly compute the dimension of the vector spaces $H^0(X,T_X),H^1(X,T_X),H^2(X,T_X)$? Here $T_X$ is the tangent sheaf of $X$.
For instance $h^0(X,T_X)$ gives the dimension of the automorphism group of $X$.
ag.algebraic-geometry sheaf-theory projective-geometry birational-geometry sheaf-cohomology
ag.algebraic-geometry sheaf-theory projective-geometry birational-geometry sheaf-cohomology
asked yesterday
user125056user125056
261
New contributor
user125056 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
user125056user125056
261
New contributor
user125056 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
user125056user125056
261
New contributor
user125056 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
user125056user125056
261
asked yesterday
user125056user125056
261
261
New contributor
user125056 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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New contributor
user125056 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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1 Answer
1
active
oldest
votes
$begingroup$
Use the normal sequence
$$
0 to T_X to T_{mathbb{P}^n}vert_X to N_{X/mathbb{P}^n} to 0,
$$
exact sequences
$$
0 to mathcal{O}_{mathbb{P}^n} to mathcal{O}_{mathbb{P}^n}(d) to i_*N_{X/mathbb{P}^n} to 0
$$
(we identify here $N_{X/mathbb{P}^n}$ with $mathcal{O}_X(d)$ and denote by $i$ the embedding $X to mathbb{P}^n$) and
$$
0 to T_{mathbb{P}^n}(-d) to T_{mathbb{P}^n} to i_*(T_{mathbb{P}^n}vert_X) to 0,
$$
and Borel-Bott-Weil Theorem to compute cohomology on $mathbb{P}^n$.
answered yesterday
SashaSasha
20.8k22755
$endgroup$
1
$begingroup$
I think that the first exact sequence you wrote holds only if $X$ is smooth. What if $X$ is singular?
$endgroup$
– user125056
yesterday
2
$begingroup$
This is right (I missed the absence of the smoothness assumption in the question). In the singular case the first sequence is not exact in the right term, but its cokernel is not so hard to control. It is isomorphic to $mathcal{O}_Z(d)$, where $Z subset mathbb{P}^n$ is the subscheme defined by ${ partial F/partial x_0 = partial F/partial x_1 = dots partial F/partial x_n = 0 }$, where $F$ is the equation of $X$.
$endgroup$
– Sasha
yesterday
$begingroup$
An addendum: this nice paper by Sernesi (arxiv.org/pdf/1306.3736.pdf) gives informations on how to control the deformations of a singular reduced hypersurface in terms of the local cohomology of the cokernel that Sasha was mentioning.
$endgroup$
– Enrico
yesterday
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Use the normal sequence
$$
0 to T_X to T_{mathbb{P}^n}vert_X to N_{X/mathbb{P}^n} to 0,
$$
exact sequences
$$
0 to mathcal{O}_{mathbb{P}^n} to mathcal{O}_{mathbb{P}^n}(d) to i_*N_{X/mathbb{P}^n} to 0
$$
(we identify here $N_{X/mathbb{P}^n}$ with $mathcal{O}_X(d)$ and denote by $i$ the embedding $X to mathbb{P}^n$) and
$$
0 to T_{mathbb{P}^n}(-d) to T_{mathbb{P}^n} to i_*(T_{mathbb{P}^n}vert_X) to 0,
$$
and Borel-Bott-Weil Theorem to compute cohomology on $mathbb{P}^n$.
answered yesterday
SashaSasha
20.8k22755
$endgroup$
1
$begingroup$
I think that the first exact sequence you wrote holds only if $X$ is smooth. What if $X$ is singular?
$endgroup$
– user125056
yesterday
2
$begingroup$
This is right (I missed the absence of the smoothness assumption in the question). In the singular case the first sequence is not exact in the right term, but its cokernel is not so hard to control. It is isomorphic to $mathcal{O}_Z(d)$, where $Z subset mathbb{P}^n$ is the subscheme defined by ${ partial F/partial x_0 = partial F/partial x_1 = dots partial F/partial x_n = 0 }$, where $F$ is the equation of $X$.
$endgroup$
– Sasha
yesterday
$begingroup$
An addendum: this nice paper by Sernesi (arxiv.org/pdf/1306.3736.pdf) gives informations on how to control the deformations of a singular reduced hypersurface in terms of the local cohomology of the cokernel that Sasha was mentioning.
$endgroup$
– Enrico
yesterday
add a comment |
$begingroup$
Use the normal sequence
$$
0 to T_X to T_{mathbb{P}^n}vert_X to N_{X/mathbb{P}^n} to 0,
$$
exact sequences
$$
0 to mathcal{O}_{mathbb{P}^n} to mathcal{O}_{mathbb{P}^n}(d) to i_*N_{X/mathbb{P}^n} to 0
$$
(we identify here $N_{X/mathbb{P}^n}$ with $mathcal{O}_X(d)$ and denote by $i$ the embedding $X to mathbb{P}^n$) and
$$
0 to T_{mathbb{P}^n}(-d) to T_{mathbb{P}^n} to i_*(T_{mathbb{P}^n}vert_X) to 0,
$$
and Borel-Bott-Weil Theorem to compute cohomology on $mathbb{P}^n$.
answered yesterday
SashaSasha
20.8k22755
$endgroup$
1
$begingroup$
I think that the first exact sequence you wrote holds only if $X$ is smooth. What if $X$ is singular?
$endgroup$
– user125056
yesterday
2
$begingroup$
This is right (I missed the absence of the smoothness assumption in the question). In the singular case the first sequence is not exact in the right term, but its cokernel is not so hard to control. It is isomorphic to $mathcal{O}_Z(d)$, where $Z subset mathbb{P}^n$ is the subscheme defined by ${ partial F/partial x_0 = partial F/partial x_1 = dots partial F/partial x_n = 0 }$, where $F$ is the equation of $X$.
$endgroup$
– Sasha
yesterday
$begingroup$
An addendum: this nice paper by Sernesi (arxiv.org/pdf/1306.3736.pdf) gives informations on how to control the deformations of a singular reduced hypersurface in terms of the local cohomology of the cokernel that Sasha was mentioning.
$endgroup$
– Enrico
yesterday
add a comment |
$begingroup$
Use the normal sequence
$$
0 to T_X to T_{mathbb{P}^n}vert_X to N_{X/mathbb{P}^n} to 0,
$$
exact sequences
$$
0 to mathcal{O}_{mathbb{P}^n} to mathcal{O}_{mathbb{P}^n}(d) to i_*N_{X/mathbb{P}^n} to 0
$$
(we identify here $N_{X/mathbb{P}^n}$ with $mathcal{O}_X(d)$ and denote by $i$ the embedding $X to mathbb{P}^n$) and
$$
0 to T_{mathbb{P}^n}(-d) to T_{mathbb{P}^n} to i_*(T_{mathbb{P}^n}vert_X) to 0,
$$
and Borel-Bott-Weil Theorem to compute cohomology on $mathbb{P}^n$.
answered yesterday
SashaSasha
20.8k22755
$endgroup$
Use the normal sequence
$$
0 to T_X to T_{mathbb{P}^n}vert_X to N_{X/mathbb{P}^n} to 0,
$$
exact sequences
$$
0 to mathcal{O}_{mathbb{P}^n} to mathcal{O}_{mathbb{P}^n}(d) to i_*N_{X/mathbb{P}^n} to 0
$$
(we identify here $N_{X/mathbb{P}^n}$ with $mathcal{O}_X(d)$ and denote by $i$ the embedding $X to mathbb{P}^n$) and
$$
0 to T_{mathbb{P}^n}(-d) to T_{mathbb{P}^n} to i_*(T_{mathbb{P}^n}vert_X) to 0,
$$
and Borel-Bott-Weil Theorem to compute cohomology on $mathbb{P}^n$.
answered yesterday
SashaSasha
20.8k22755
answered yesterday
SashaSasha
20.8k22755
answered yesterday
SashaSasha
20.8k22755
answered yesterday
SashaSasha
20.8k22755
20.8k22755
1
$begingroup$
I think that the first exact sequence you wrote holds only if $X$ is smooth. What if $X$ is singular?
$endgroup$
– user125056
yesterday
2
$begingroup$
This is right (I missed the absence of the smoothness assumption in the question). In the singular case the first sequence is not exact in the right term, but its cokernel is not so hard to control. It is isomorphic to $mathcal{O}_Z(d)$, where $Z subset mathbb{P}^n$ is the subscheme defined by ${ partial F/partial x_0 = partial F/partial x_1 = dots partial F/partial x_n = 0 }$, where $F$ is the equation of $X$.
$endgroup$
– Sasha
yesterday
$begingroup$
An addendum: this nice paper by Sernesi (arxiv.org/pdf/1306.3736.pdf) gives informations on how to control the deformations of a singular reduced hypersurface in terms of the local cohomology of the cokernel that Sasha was mentioning.
$endgroup$
– Enrico
yesterday
add a comment |
1
$begingroup$
I think that the first exact sequence you wrote holds only if $X$ is smooth. What if $X$ is singular?
$endgroup$
– user125056
yesterday
2
$begingroup$
This is right (I missed the absence of the smoothness assumption in the question). In the singular case the first sequence is not exact in the right term, but its cokernel is not so hard to control. It is isomorphic to $mathcal{O}_Z(d)$, where $Z subset mathbb{P}^n$ is the subscheme defined by ${ partial F/partial x_0 = partial F/partial x_1 = dots partial F/partial x_n = 0 }$, where $F$ is the equation of $X$.
$endgroup$
– Sasha
yesterday
$begingroup$
An addendum: this nice paper by Sernesi (arxiv.org/pdf/1306.3736.pdf) gives informations on how to control the deformations of a singular reduced hypersurface in terms of the local cohomology of the cokernel that Sasha was mentioning.
$endgroup$
– Enrico
yesterday
1
1
$begingroup$
I think that the first exact sequence you wrote holds only if $X$ is smooth. What if $X$ is singular?
$endgroup$
– user125056
yesterday
$begingroup$
I think that the first exact sequence you wrote holds only if $X$ is smooth. What if $X$ is singular?
$endgroup$
– user125056
yesterday
2
2
$begingroup$
This is right (I missed the absence of the smoothness assumption in the question). In the singular case the first sequence is not exact in the right term, but its cokernel is not so hard to control. It is isomorphic to $mathcal{O}_Z(d)$, where $Z subset mathbb{P}^n$ is the subscheme defined by ${ partial F/partial x_0 = partial F/partial x_1 = dots partial F/partial x_n = 0 }$, where $F$ is the equation of $X$.
$endgroup$
– Sasha
yesterday
$begingroup$
This is right (I missed the absence of the smoothness assumption in the question). In the singular case the first sequence is not exact in the right term, but its cokernel is not so hard to control. It is isomorphic to $mathcal{O}_Z(d)$, where $Z subset mathbb{P}^n$ is the subscheme defined by ${ partial F/partial x_0 = partial F/partial x_1 = dots partial F/partial x_n = 0 }$, where $F$ is the equation of $X$.
$endgroup$
– Sasha
yesterday
$begingroup$
An addendum: this nice paper by Sernesi (arxiv.org/pdf/1306.3736.pdf) gives informations on how to control the deformations of a singular reduced hypersurface in terms of the local cohomology of the cokernel that Sasha was mentioning.
$endgroup$
– Enrico
yesterday
$begingroup$
An addendum: this nice paper by Sernesi (arxiv.org/pdf/1306.3736.pdf) gives informations on how to control the deformations of a singular reduced hypersurface in terms of the local cohomology of the cokernel that Sasha was mentioning.
$endgroup$
– Enrico
yesterday
add a comment |
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