Reference request: most hyperplane sections contain one node

Question

Let $X\subset \mathbb {CP}^n$ 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.

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.