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In Voisin book "Hodge theory and Complex Algebraic Geometry 2". There is the following corollary

Corollary 2.10. If $X\subset \mathbb{P}^N$ is a smooth projective complex variety, then a generic pencil $(X_t)_{t\in \mathbb{P}^1 }$ of hyperplane sections of $X$ is a Lefschetz pencil.

Then, there is no more information about "generic pencil" in the book. I would like to know how a generic pencil is defined.

Thank you!

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    $\begingroup$ Intuitively, the set of pencils of hyperplanes such that all tangent hyperplanes in the pencil have the simplest possible tangent behavior form a non-empty open subset in the variety parametrizing all linear pencils. $\endgroup$
    – Francesco Polizzi
    Commented May 24, 2021 at 8:34

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The set of codimension two linear subspaces of $\mathbb{P}^N$ is parameterized by a Grassmanian, say $G$. If $L\in G$, then the set of hyperplanes containing $L$ gives rise to a pencil of hyperplane sections $\{X_t\}$. Generic, in this context, means that there is a nonempty Zariski open $U\subset G$, such that the corresponding pencil is Lefschetz. I don't have Voisin's book with me at the moment, but I imagine she would give more precise information about what $U$, or perhaps $G-U$, looks like.

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