# Given a smooth hyperplane section Y of a variety X there exists a Lefschetz pencil of hyperplane sections of X containing Y

Let $$X$$ be a variety contained in $$\mathbb{P}^N$$ and let $$Y$$ be a smooth hyperplane section of $$X$$. I have read in page 54 of Voisin's book "Hodge theory and complex algebraic geometry II" the following:

there exists a Lefschetz pencil of hyperplane sections of $$X$$ of which $$Y$$ is one member.

I would like to know why this is true. Any reference for it will be also very helpful. Thank you very much :)!

A pencil of hyperplane sections of $$X$$ corresponds to a line in the dual projective space $$\check{\mathbb{P}}^N$$. It is a Lefschetz pencil if and only if the line is transverse to the projectively dual variety $$\check{X} \subset \check{\mathbb{P}}^N.$$ So, a restatement of your claim is that for a point $$y \in \check{\mathbb{P}}^N$$ not on $$\check{X}$$ there is a line through $$y$$ transverse to $$\check{X}$$, which is obvious --- just consider the linear projection map $$\pi_y \colon \check{X} \to \check{\mathbb{P}}^{N-1}$$ from $$y$$, choose a point in the target not on the critical locus of $$\pi_y$$, and take the corresponding line.
• Dear Sasha thank you very much for your answer!... could you tell me how $\pi_y$ is defined and what you mean for the critical locus of $\pi_y$? Apr 2 at 10:30
• $\pi_y$ is the linear projection $\mathbb{P}^N \dashrightarrow \mathbb{P}^{N-1}$ with center at $y$, restricted to $\check{X}$. The critical locus, or rather the set of critical values, is the image of the union of the singular locus of $\check{X}$ and the set of points of $\check{X}$ at which the differential of $\pi_y$ is not surjective. Apr 2 at 11:52
• Dear Sasha thank you very much! for the linear projection $\pi_y$ with center in $y$, you mean a notion similar to the Stereographic projection? could you suggest a reference where I can read what exactly you mean :)... Apr 2 at 14:07
• Choose coordinates such that $y = (1:0:\dots:0)$. Then $$\pi_y(x_0:x_1:\dots:x_n) = (x_1:\dots:x_n).$$ Apr 2 at 14:56
• Dear Sasha thank you very much for the clarifications :)!... I have the following questions: 1). Why I need to choose a point in the target not on the critical locus of $\pi_y$? Apr 4 at 21:59