Cohomology of tangent sheaf of a hypersurface
$begingroup$
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 11 hours ago
user125056user125056
161
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$begingroup$
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 11 hours ago
user125056user125056
161
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 11 hours ago
user125056user125056
161
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 11 hours ago
user125056user125056
161
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 11 hours ago
user125056user125056
161
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 11 hours ago
user125056user125056
161
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user125056 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 11 hours ago
user125056user125056
161
asked 11 hours ago
user125056user125056
161
161
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.
New contributor
user125056 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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add a comment |
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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 11 hours ago
SashaSasha
20.8k22655
$endgroup$
1
$begingroup$
I think that the first exact sequence you wrote holds only if $X$ is smooth. What if $X$ is singular?
$endgroup$
– user125056
10 hours ago
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
9 hours ago
$begingroup$
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
6 hours ago
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 11 hours ago
SashaSasha
20.8k22655
$endgroup$
1
$begingroup$
I think that the first exact sequence you wrote holds only if $X$ is smooth. What if $X$ is singular?
$endgroup$
– user125056
10 hours ago
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
9 hours ago
$begingroup$
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
6 hours ago
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 11 hours ago
SashaSasha
20.8k22655
$endgroup$
1
$begingroup$
I think that the first exact sequence you wrote holds only if $X$ is smooth. What if $X$ is singular?
$endgroup$
– user125056
10 hours ago
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
9 hours ago
$begingroup$
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
6 hours ago
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 11 hours ago
SashaSasha
20.8k22655
$endgroup$
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 11 hours ago
SashaSasha
20.8k22655
answered 11 hours ago
SashaSasha
20.8k22655
answered 11 hours ago
SashaSasha
20.8k22655
answered 11 hours ago
SashaSasha
20.8k22655
20.8k22655
1
$begingroup$
I think that the first exact sequence you wrote holds only if $X$ is smooth. What if $X$ is singular?
$endgroup$
– user125056
10 hours ago
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
9 hours ago
$begingroup$
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
6 hours ago
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
10 hours ago
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
9 hours ago
$begingroup$
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
6 hours ago
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
10 hours ago
$begingroup$
I think that the first exact sequence you wrote holds only if $X$ is smooth. What if $X$ is singular?
$endgroup$
– user125056
10 hours ago
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
9 hours ago
$begingroup$
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
9 hours ago
$begingroup$
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
6 hours ago
$begingroup$
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
6 hours ago
add a comment |
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