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Let $X\subset \mathbb P^n$ be a smooth hypersurface over base field $\mathbb C$. A pencil of hyperplanes is just a projective line $(X_t)$ in $\mathbb P^{n*}$. It is called a Lefschetz pencil if it satisfies the following: (I followed the definition in SGA)

(i) The axis (intersection of $X_t$'s) intersects with $X$ transversally.

(ii) Almost all (enough for one) $X_t$ intersects with $X$ transversally.

(iii) The non-transversal intersection in (ii) contains one node.

And on SGA it also proves that almost all pencils are Lefschetz. Now I want to know how many are those non-Lefschetz pencils, more precisely:

Does there exist some $X$, on which all the non-Lefschetz pencils cover the $\mathbb P^{n*}$? i.e. for any hyperplane section $H$, we can find non-Lefschetz pencil $(X_t)$ pass through it.

I guess it does not exist, but have no idea how to prove it. Any hints or references would be helpful.

Remark

I should remove the condition (i) otherwise it is trivial.

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Assume there is a single hyperplane section $X_0$ of $X$ which has worse singularity than just a node. Then any pencil containing $X_0$ is non-Lefschetz, and these pencils cover all the $\mathbb{P}^{n*}$.

Of course, if there is no such $X_0$ as above, this construction does not work. However, I think the only case when such $X_0$ does not exist is that of $X$ being a smooth quadric. In that case the projective dual variety to $X$ is yet another smooth quadric, and lines in $\mathbb{P}^{n*}$ tangent to it provide non-Lefschetz pencils that cover all the $\mathbb{P}^{n*}$.

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  • $\begingroup$ Thanks for your answer. Btw can I ask another question (maybe I should start a new question but anyway) why we need the condition (i) in the definition? Isn’t it followed by (ii)? $\endgroup$ – User X Nov 17 '18 at 9:39
  • $\begingroup$ Please, give a precise reference to the definition. $\endgroup$ – Sasha Nov 17 '18 at 10:12
  • $\begingroup$ I followed the definition in Groupes de monodromie en géométrie algébrique. II. Lecture Notes in Mathematics, Vol. 340. Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II). $\endgroup$ – User X Nov 17 '18 at 11:20
  • $\begingroup$ A precise reference should include expose and section numbers. In fact "axis" ("l'axe") in English is usually called "base locus". It surely doesn't follow from (ii). For instance, take $X \subset \mathbb{P}^2$ to be a smooth conic, and the pencil of lines through a point ON $X$. $\endgroup$ – Sasha Nov 17 '18 at 14:44
  • $\begingroup$ Sorry I thought something wrong. Thanks. $\endgroup$ – User X Nov 17 '18 at 19:07

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