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I hear sometimes about a generic hyperplane of a projective space over $\mathbb{C}$, for example in the Noether-Lefschetz theorem. What is the definition of it?

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Saying that something is true for a generic hyperplane in $\mathbf P^n$ is saying that it's true for all hyperplanes outside of a proper subvariety of the space of hyperplanes (which is also a projective space of dimension $n$). I think it can also mean, more generally, for all hyperplanes outside of a countable union of proper subvarieties.

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