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I would like to understand the the idea behind the historically original proof by Lefschetz of his Hyperplane theorem sketched roughly Here. The basic setup:

Let $X$ be an $n$ -dimensional complex projective algebraic variety closed embedded in $\mathbb{CP}^n $ and let $Y$ be a hyperplane section of $X$.

Lefschetz used his idea of a Lefschetz pencil to prove the theorem. Rather than considering the hyperplane section $Y$ alone, he put it into a family of hyperplane sections $Y_t$, where $Y=Y_0$. Because a generic hyperplane section is smooth, all but a finite number of $Y_t$ are smooth varieties. After removing these points from the $t$ -plane and making an additional finite number of slits, the resulting family of hyperplane sections is topologically trivial. That is, it is a product of a generic $Y_t$ with an open subset of the $t$-plane. In modern terns the $t$-plane is nothing but the complex plane $\mathbb{P}^1$ and we have a canonical rational map from the union of the hypersections $Y_t$ to $\mathbb{P}^1$. $X$, therefore, can be understood if one understands how hyperplane sections are identified across the slits and at the singular points. Away from the singular points, the identification can be described inductively. At the singular points, the Morse lemma implies that there is a choice of coordinate system for X of a particularly simple form. This coordinate system can be used to prove the theorem directly.

Question: I not understand the part where one allows "inductively" to glue/ realize identifications "away from singular points inductively." What is the precise statement over which we perform this "induction" and over which quantity? Subspace dimension, ie like in a Noetherian induction? This is important in order to understand which identifications is here "legal" to assume as be already done in "induction hypothesis" .

Could somebody spell out the precise "statement" on which we do this induction the proof is referring to? I'm confused namely about what is allowed to be assumed to be already done by induction hypothesis, and what not, and why?

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    $\begingroup$ This seems more a math question than a history question $\endgroup$ Aug 11, 2023 at 18:17

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