Reference request: most hyperplane sections contain one node
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Let $Xsubset mathbb {CP}^n$ be a smooth hypersurface. It is known that most of hyperplane sections are smooth. My question is
Is it true that most of the singular sections contain one node?
I think it is true and also used it for a long time, but I just realized I never knew how to prove this. Could someone give a reference about it?
I also tried to do computation directly. For hypersurfaces of degree $2$, they are all of the form like $x^2+y^2+z^2+w^2=0$, so easy to see all the sections are smooth. For higher degree I don't know a good way to do it.
algebraic-geometry reference-request complex-geometry
asked Nov 18 at 16:55
Akatsuki
9451624
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show 6 more comments
up vote
0
down vote
favorite
Let $Xsubset mathbb {CP}^n$ be a smooth hypersurface. It is known that most of hyperplane sections are smooth. My question is
Is it true that most of the singular sections contain one node?
I think it is true and also used it for a long time, but I just realized I never knew how to prove this. Could someone give a reference about it?
I also tried to do computation directly. For hypersurfaces of degree $2$, they are all of the form like $x^2+y^2+z^2+w^2=0$, so easy to see all the sections are smooth. For higher degree I don't know a good way to do it.
algebraic-geometry reference-request complex-geometry
asked Nov 18 at 16:55
Akatsuki
9451624
I don't understand what do you mean for hypersurface of degree $2$ : for example the intersection of $x_0 = 0$ with $x_0^2 + x_1^2 + dots + x_n^2 = 0$ gives you a singular quadric.
– Nicolas Hemelsoet
Nov 18 at 17:12
It's always possible to find a pencil of hyperplane sections such that the singular sections contain at worst one node. These are called Lefschetz pencils. Is that what you're looking for?
– Samir Canning
Nov 18 at 18:50
@NicolasHemelsoet Isn't the intersection still smooth? We are considering the homogenous coordinate.
– Akatsuki
Nov 18 at 19:19
@Akatsuki : no since there is one less variable, e.g $x_1^2 + x_2^2 = 0$ is not smooth in $Bbb P^2$ since it's a union of two lines intersecting at a point. In general, quadrics in $Bbb P^n$ are classified by their rank, and smooth only if the rank is maximal.
– Nicolas Hemelsoet
Nov 18 at 19:22
Also I have no ideas for a reference sorry, but maybe the first pages of Lamotke's article about topology of projective varieties contains something ?
– Nicolas Hemelsoet
Nov 18 at 19:24
|
show 6 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $Xsubset mathbb {CP}^n$ be a smooth hypersurface. It is known that most of hyperplane sections are smooth. My question is
Is it true that most of the singular sections contain one node?
I think it is true and also used it for a long time, but I just realized I never knew how to prove this. Could someone give a reference about it?
I also tried to do computation directly. For hypersurfaces of degree $2$, they are all of the form like $x^2+y^2+z^2+w^2=0$, so easy to see all the sections are smooth. For higher degree I don't know a good way to do it.
algebraic-geometry reference-request complex-geometry
asked Nov 18 at 16:55
Akatsuki
9451624
Let $Xsubset mathbb {CP}^n$ be a smooth hypersurface. It is known that most of hyperplane sections are smooth. My question is
Is it true that most of the singular sections contain one node?
I think it is true and also used it for a long time, but I just realized I never knew how to prove this. Could someone give a reference about it?
I also tried to do computation directly. For hypersurfaces of degree $2$, they are all of the form like $x^2+y^2+z^2+w^2=0$, so easy to see all the sections are smooth. For higher degree I don't know a good way to do it.
algebraic-geometry reference-request complex-geometry
algebraic-geometry reference-request complex-geometry
asked Nov 18 at 16:55
Akatsuki
9451624
asked Nov 18 at 16:55
Akatsuki
9451624
asked Nov 18 at 16:55
Akatsuki
9451624
asked Nov 18 at 16:55
Akatsuki
9451624
asked Nov 18 at 16:55
Akatsuki
9451624
9451624
I don't understand what do you mean for hypersurface of degree $2$ : for example the intersection of $x_0 = 0$ with $x_0^2 + x_1^2 + dots + x_n^2 = 0$ gives you a singular quadric.
– Nicolas Hemelsoet
Nov 18 at 17:12
It's always possible to find a pencil of hyperplane sections such that the singular sections contain at worst one node. These are called Lefschetz pencils. Is that what you're looking for?
– Samir Canning
Nov 18 at 18:50
@NicolasHemelsoet Isn't the intersection still smooth? We are considering the homogenous coordinate.
– Akatsuki
Nov 18 at 19:19
@Akatsuki : no since there is one less variable, e.g $x_1^2 + x_2^2 = 0$ is not smooth in $Bbb P^2$ since it's a union of two lines intersecting at a point. In general, quadrics in $Bbb P^n$ are classified by their rank, and smooth only if the rank is maximal.
– Nicolas Hemelsoet
Nov 18 at 19:22
Also I have no ideas for a reference sorry, but maybe the first pages of Lamotke's article about topology of projective varieties contains something ?
– Nicolas Hemelsoet
Nov 18 at 19:24
|
show 6 more comments
I don't understand what do you mean for hypersurface of degree $2$ : for example the intersection of $x_0 = 0$ with $x_0^2 + x_1^2 + dots + x_n^2 = 0$ gives you a singular quadric.
– Nicolas Hemelsoet
Nov 18 at 17:12
It's always possible to find a pencil of hyperplane sections such that the singular sections contain at worst one node. These are called Lefschetz pencils. Is that what you're looking for?
– Samir Canning
Nov 18 at 18:50
@NicolasHemelsoet Isn't the intersection still smooth? We are considering the homogenous coordinate.
– Akatsuki
Nov 18 at 19:19
@Akatsuki : no since there is one less variable, e.g $x_1^2 + x_2^2 = 0$ is not smooth in $Bbb P^2$ since it's a union of two lines intersecting at a point. In general, quadrics in $Bbb P^n$ are classified by their rank, and smooth only if the rank is maximal.
– Nicolas Hemelsoet
Nov 18 at 19:22
Also I have no ideas for a reference sorry, but maybe the first pages of Lamotke's article about topology of projective varieties contains something ?
– Nicolas Hemelsoet
Nov 18 at 19:24
I don't understand what do you mean for hypersurface of degree $2$ : for example the intersection of $x_0 = 0$ with $x_0^2 + x_1^2 + dots + x_n^2 = 0$ gives you a singular quadric.
– Nicolas Hemelsoet
Nov 18 at 17:12
I don't understand what do you mean for hypersurface of degree $2$ : for example the intersection of $x_0 = 0$ with $x_0^2 + x_1^2 + dots + x_n^2 = 0$ gives you a singular quadric.
– Nicolas Hemelsoet
Nov 18 at 17:12
It's always possible to find a pencil of hyperplane sections such that the singular sections contain at worst one node. These are called Lefschetz pencils. Is that what you're looking for?
– Samir Canning
Nov 18 at 18:50
It's always possible to find a pencil of hyperplane sections such that the singular sections contain at worst one node. These are called Lefschetz pencils. Is that what you're looking for?
– Samir Canning
Nov 18 at 18:50
@NicolasHemelsoet Isn't the intersection still smooth? We are considering the homogenous coordinate.
– Akatsuki
Nov 18 at 19:19
@NicolasHemelsoet Isn't the intersection still smooth? We are considering the homogenous coordinate.
– Akatsuki
Nov 18 at 19:19
@Akatsuki : no since there is one less variable, e.g $x_1^2 + x_2^2 = 0$ is not smooth in $Bbb P^2$ since it's a union of two lines intersecting at a point. In general, quadrics in $Bbb P^n$ are classified by their rank, and smooth only if the rank is maximal.
– Nicolas Hemelsoet
Nov 18 at 19:22
@Akatsuki : no since there is one less variable, e.g $x_1^2 + x_2^2 = 0$ is not smooth in $Bbb P^2$ since it's a union of two lines intersecting at a point. In general, quadrics in $Bbb P^n$ are classified by their rank, and smooth only if the rank is maximal.
– Nicolas Hemelsoet
Nov 18 at 19:22
Also I have no ideas for a reference sorry, but maybe the first pages of Lamotke's article about topology of projective varieties contains something ?
– Nicolas Hemelsoet
Nov 18 at 19:24
Also I have no ideas for a reference sorry, but maybe the first pages of Lamotke's article about topology of projective varieties contains something ?
– Nicolas Hemelsoet
Nov 18 at 19:24
|
show 6 more comments
1 Answer
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Proposition : Let $hat{X}$ be the dual variety. Let $T subset hat{Bbb P^n}$ be a line such that $T$ avoids the singular locus of $hat{X}$ and is transverse to $hat{X}$ : then $H_t cap X$ is a Lefschetz pencil.
Clearly, this implies that if $t$ is not in the singular locus of $hat{X}$ then $H_t cap X$ has a nodal singularity. Since the singular hyperplanes sections are parametrized by $hat{X}$ (also proved in Lamotke) you obtain that a generic singular hyperplane section has nodal singularities.
For a proof of the proposition, see "The topology of projective algebraic varieties after S.Lefschetz" by K. Lamotke, paragraph 1.6, in particular 1.6.4.
answered Nov 20 at 22:49
Nicolas Hemelsoet
5,5552417
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Proposition : Let $hat{X}$ be the dual variety. Let $T subset hat{Bbb P^n}$ be a line such that $T$ avoids the singular locus of $hat{X}$ and is transverse to $hat{X}$ : then $H_t cap X$ is a Lefschetz pencil.
Clearly, this implies that if $t$ is not in the singular locus of $hat{X}$ then $H_t cap X$ has a nodal singularity. Since the singular hyperplanes sections are parametrized by $hat{X}$ (also proved in Lamotke) you obtain that a generic singular hyperplane section has nodal singularities.
For a proof of the proposition, see "The topology of projective algebraic varieties after S.Lefschetz" by K. Lamotke, paragraph 1.6, in particular 1.6.4.
answered Nov 20 at 22:49
Nicolas Hemelsoet
5,5552417
add a comment |
up vote
0
down vote
Proposition : Let $hat{X}$ be the dual variety. Let $T subset hat{Bbb P^n}$ be a line such that $T$ avoids the singular locus of $hat{X}$ and is transverse to $hat{X}$ : then $H_t cap X$ is a Lefschetz pencil.
Clearly, this implies that if $t$ is not in the singular locus of $hat{X}$ then $H_t cap X$ has a nodal singularity. Since the singular hyperplanes sections are parametrized by $hat{X}$ (also proved in Lamotke) you obtain that a generic singular hyperplane section has nodal singularities.
For a proof of the proposition, see "The topology of projective algebraic varieties after S.Lefschetz" by K. Lamotke, paragraph 1.6, in particular 1.6.4.
answered Nov 20 at 22:49
Nicolas Hemelsoet
5,5552417
add a comment |
up vote
0
down vote
up vote
0
down vote
Proposition : Let $hat{X}$ be the dual variety. Let $T subset hat{Bbb P^n}$ be a line such that $T$ avoids the singular locus of $hat{X}$ and is transverse to $hat{X}$ : then $H_t cap X$ is a Lefschetz pencil.
Clearly, this implies that if $t$ is not in the singular locus of $hat{X}$ then $H_t cap X$ has a nodal singularity. Since the singular hyperplanes sections are parametrized by $hat{X}$ (also proved in Lamotke) you obtain that a generic singular hyperplane section has nodal singularities.
For a proof of the proposition, see "The topology of projective algebraic varieties after S.Lefschetz" by K. Lamotke, paragraph 1.6, in particular 1.6.4.
answered Nov 20 at 22:49
Nicolas Hemelsoet
5,5552417
Proposition : Let $hat{X}$ be the dual variety. Let $T subset hat{Bbb P^n}$ be a line such that $T$ avoids the singular locus of $hat{X}$ and is transverse to $hat{X}$ : then $H_t cap X$ is a Lefschetz pencil.
Clearly, this implies that if $t$ is not in the singular locus of $hat{X}$ then $H_t cap X$ has a nodal singularity. Since the singular hyperplanes sections are parametrized by $hat{X}$ (also proved in Lamotke) you obtain that a generic singular hyperplane section has nodal singularities.
For a proof of the proposition, see "The topology of projective algebraic varieties after S.Lefschetz" by K. Lamotke, paragraph 1.6, in particular 1.6.4.
answered Nov 20 at 22:49
Nicolas Hemelsoet
5,5552417
answered Nov 20 at 22:49
Nicolas Hemelsoet
5,5552417
answered Nov 20 at 22:49
Nicolas Hemelsoet
5,5552417
answered Nov 20 at 22:49
Nicolas Hemelsoet
5,5552417
5,5552417
add a comment |
add a comment |
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I don't understand what do you mean for hypersurface of degree $2$ : for example the intersection of $x_0 = 0$ with $x_0^2 + x_1^2 + dots + x_n^2 = 0$ gives you a singular quadric.
– Nicolas Hemelsoet
Nov 18 at 17:12
It's always possible to find a pencil of hyperplane sections such that the singular sections contain at worst one node. These are called Lefschetz pencils. Is that what you're looking for?
– Samir Canning
Nov 18 at 18:50
@NicolasHemelsoet Isn't the intersection still smooth? We are considering the homogenous coordinate.
– Akatsuki
Nov 18 at 19:19
@Akatsuki : no since there is one less variable, e.g $x_1^2 + x_2^2 = 0$ is not smooth in $Bbb P^2$ since it's a union of two lines intersecting at a point. In general, quadrics in $Bbb P^n$ are classified by their rank, and smooth only if the rank is maximal.
– Nicolas Hemelsoet
Nov 18 at 19:22
Also I have no ideas for a reference sorry, but maybe the first pages of Lamotke's article about topology of projective varieties contains something ?
– Nicolas Hemelsoet
Nov 18 at 19:24