canonical bundle of Veronese embedding The Next CEO of Stack OverflowCanonical Vector Bundle associated to a complete intersectionHartshorne proof of adjunction formula proposition II.8.20Normal bundle to complete intersection in $mathbbP^n$Complete Intersection does not lie in a hyperplane?Canonical bundle of the Lagrangian GrassmannianCanonical divisor of smooth complete intersection curve is very ample for genus $g geq 2$.When is the canonical sheaf of a curve very ample?Pullback of line bundle of negative degree has no global sections.Proof of Generalized Bézout theorem using Veronese Embedding mapCanonical bundle of blow up at singular point
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canonical bundle of Veronese embedding
The Next CEO of Stack OverflowCanonical Vector Bundle associated to a complete intersectionHartshorne proof of adjunction formula proposition II.8.20Normal bundle to complete intersection in $mathbbP^n$Complete Intersection does not lie in a hyperplane?Canonical bundle of the Lagrangian GrassmannianCanonical divisor of smooth complete intersection curve is very ample for genus $g geq 2$.When is the canonical sheaf of a curve very ample?Pullback of line bundle of negative degree has no global sections.Proof of Generalized Bézout theorem using Veronese Embedding mapCanonical bundle of blow up at singular point
$begingroup$
Suppose we are given a complete intersection $X$ of codimension $r=n-d$ in $mathbbP^n$ where the degrees of the hypersurfaces are $d_i$ and $d$ is the dimension of $X$. Then the canonical bundle $omega_X=mathcalO_X(sum d_i-n-1)$. Now suppose we embedd $X$ via a Veronese embdding in some $mathbbP^N$ for some $N>>0$ so that one has inclusion $i:Xrightarrow mathbbP^N$. By Hartshorne Proposition 8.20 one can calculate $omega_X$ is this case but what is the normal bundle $mathcalN_X/mathbbP^N$ in this case?
algebraic-geometry
asked Dec 27 '13 at 13:55
user109227user109227
28916
$endgroup$
add a comment |
$begingroup$
Suppose we are given a complete intersection $X$ of codimension $r=n-d$ in $mathbbP^n$ where the degrees of the hypersurfaces are $d_i$ and $d$ is the dimension of $X$. Then the canonical bundle $omega_X=mathcalO_X(sum d_i-n-1)$. Now suppose we embedd $X$ via a Veronese embdding in some $mathbbP^N$ for some $N>>0$ so that one has inclusion $i:Xrightarrow mathbbP^N$. By Hartshorne Proposition 8.20 one can calculate $omega_X$ is this case but what is the normal bundle $mathcalN_X/mathbbP^N$ in this case?
algebraic-geometry
asked Dec 27 '13 at 13:55
user109227user109227
28916
$endgroup$
add a comment |
$begingroup$
Suppose we are given a complete intersection $X$ of codimension $r=n-d$ in $mathbbP^n$ where the degrees of the hypersurfaces are $d_i$ and $d$ is the dimension of $X$. Then the canonical bundle $omega_X=mathcalO_X(sum d_i-n-1)$. Now suppose we embedd $X$ via a Veronese embdding in some $mathbbP^N$ for some $N>>0$ so that one has inclusion $i:Xrightarrow mathbbP^N$. By Hartshorne Proposition 8.20 one can calculate $omega_X$ is this case but what is the normal bundle $mathcalN_X/mathbbP^N$ in this case?
algebraic-geometry
asked Dec 27 '13 at 13:55
user109227user109227
28916
$endgroup$
Suppose we are given a complete intersection $X$ of codimension $r=n-d$ in $mathbbP^n$ where the degrees of the hypersurfaces are $d_i$ and $d$ is the dimension of $X$. Then the canonical bundle $omega_X=mathcalO_X(sum d_i-n-1)$. Now suppose we embedd $X$ via a Veronese embdding in some $mathbbP^N$ for some $N>>0$ so that one has inclusion $i:Xrightarrow mathbbP^N$. By Hartshorne Proposition 8.20 one can calculate $omega_X$ is this case but what is the normal bundle $mathcalN_X/mathbbP^N$ in this case?
algebraic-geometry
algebraic-geometry
asked Dec 27 '13 at 13:55
user109227user109227
28916
asked Dec 27 '13 at 13:55
user109227user109227
28916
asked Dec 27 '13 at 13:55
user109227user109227
28916
asked Dec 27 '13 at 13:55
user109227user109227
28916
asked Dec 27 '13 at 13:55
user109227user109227
28916
28916
add a comment |
add a comment |
1 Answer
1
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oldest
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$begingroup$
Suppose $M$ subset of $Bbb R^m$ is a closed embedded submanifold. if $M$ admits a global defining function, show that is normal bundle is trivial.
Conversely, if $M$ has trivial normal bundle, show that there is a nighborhood $U$ of $M$ in $Bbb R^n$ and a submersion $K$ from $U$ to $Bbb R^k$ such that $M=K^-1(U)$.
edited Mar 18 at 22:07
Yanior Weg
2,73211346
answered Jan 31 '16 at 9:44
AliAli
11
$endgroup$
$begingroup$
The question is about projective algebraic geometry (over an algebraically closed field, I suppose) What you wrote does not fit that. Already a conic in the projective plane has a non-trivial normal bundle in this context.
$endgroup$
– Mariano Suárez-Álvarez
Jan 31 '16 at 10:45
add a comment |
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1 Answer
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1 Answer
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active
oldest
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active
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$begingroup$
Suppose $M$ subset of $Bbb R^m$ is a closed embedded submanifold. if $M$ admits a global defining function, show that is normal bundle is trivial.
Conversely, if $M$ has trivial normal bundle, show that there is a nighborhood $U$ of $M$ in $Bbb R^n$ and a submersion $K$ from $U$ to $Bbb R^k$ such that $M=K^-1(U)$.
edited Mar 18 at 22:07
Yanior Weg
2,73211346
answered Jan 31 '16 at 9:44
AliAli
11
$endgroup$
$begingroup$
The question is about projective algebraic geometry (over an algebraically closed field, I suppose) What you wrote does not fit that. Already a conic in the projective plane has a non-trivial normal bundle in this context.
$endgroup$
– Mariano Suárez-Álvarez
Jan 31 '16 at 10:45
add a comment |
$begingroup$
Suppose $M$ subset of $Bbb R^m$ is a closed embedded submanifold. if $M$ admits a global defining function, show that is normal bundle is trivial.
Conversely, if $M$ has trivial normal bundle, show that there is a nighborhood $U$ of $M$ in $Bbb R^n$ and a submersion $K$ from $U$ to $Bbb R^k$ such that $M=K^-1(U)$.
edited Mar 18 at 22:07
Yanior Weg
2,73211346
answered Jan 31 '16 at 9:44
AliAli
11
$endgroup$
$begingroup$
The question is about projective algebraic geometry (over an algebraically closed field, I suppose) What you wrote does not fit that. Already a conic in the projective plane has a non-trivial normal bundle in this context.
$endgroup$
– Mariano Suárez-Álvarez
Jan 31 '16 at 10:45
add a comment |
$begingroup$
Suppose $M$ subset of $Bbb R^m$ is a closed embedded submanifold. if $M$ admits a global defining function, show that is normal bundle is trivial.
Conversely, if $M$ has trivial normal bundle, show that there is a nighborhood $U$ of $M$ in $Bbb R^n$ and a submersion $K$ from $U$ to $Bbb R^k$ such that $M=K^-1(U)$.
edited Mar 18 at 22:07
Yanior Weg
2,73211346
answered Jan 31 '16 at 9:44
AliAli
11
$endgroup$
Suppose $M$ subset of $Bbb R^m$ is a closed embedded submanifold. if $M$ admits a global defining function, show that is normal bundle is trivial.
Conversely, if $M$ has trivial normal bundle, show that there is a nighborhood $U$ of $M$ in $Bbb R^n$ and a submersion $K$ from $U$ to $Bbb R^k$ such that $M=K^-1(U)$.
edited Mar 18 at 22:07
Yanior Weg
2,73211346
answered Jan 31 '16 at 9:44
AliAli
11
edited Mar 18 at 22:07
Yanior Weg
2,73211346
edited Mar 18 at 22:07
Yanior Weg
2,73211346
edited Mar 18 at 22:07
Yanior Weg
2,73211346
2,73211346
answered Jan 31 '16 at 9:44
AliAli
11
answered Jan 31 '16 at 9:44
AliAli
11
answered Jan 31 '16 at 9:44
AliAli
11
11
$begingroup$
The question is about projective algebraic geometry (over an algebraically closed field, I suppose) What you wrote does not fit that. Already a conic in the projective plane has a non-trivial normal bundle in this context.
$endgroup$
– Mariano Suárez-Álvarez
Jan 31 '16 at 10:45
add a comment |
$begingroup$
The question is about projective algebraic geometry (over an algebraically closed field, I suppose) What you wrote does not fit that. Already a conic in the projective plane has a non-trivial normal bundle in this context.
$endgroup$
– Mariano Suárez-Álvarez
Jan 31 '16 at 10:45
$begingroup$
The question is about projective algebraic geometry (over an algebraically closed field, I suppose) What you wrote does not fit that. Already a conic in the projective plane has a non-trivial normal bundle in this context.
$endgroup$
– Mariano Suárez-Álvarez
Jan 31 '16 at 10:45
$begingroup$
The question is about projective algebraic geometry (over an algebraically closed field, I suppose) What you wrote does not fit that. Already a conic in the projective plane has a non-trivial normal bundle in this context.
$endgroup$
– Mariano Suárez-Álvarez
Jan 31 '16 at 10:45
add a comment |
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