Reference request: most hyperplane sections contain one node












0














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







share|cite|improve this question






















  • 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
















0














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







share|cite|improve this question






















  • 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














0












0








0







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







share|cite|improve this question













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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 18 at 16:55









Akatsuki

9771724




9771724












  • 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










  • 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










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.






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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.






    share|cite|improve this answer


























      1















      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.






      share|cite|improve this answer
























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        1







        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.






        share|cite|improve this answer













        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.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 20 at 22:49









        Nicolas Hemelsoet

        5,7452417




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