I'm trying to understand the main ideas used in the original proof by Lefschetz of his Hyperplane theorem. Here it is sketched shortly (source: Here) and I want to fill the gaps:
Let $X$ be an $n$-dimensional complex projective algebraic variety closed embedded in $\mathbb{CP}^n$ and let $Y$ be a hyperplane section of $X$ such that $U = X ∖ Y$ is smooth.
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. $X$, 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.
The original source is 'L'Analysis situs et la géométrie algébrique' but I can't get along with that text. So I would be happy if someone could help me to clear up the understanding problems.
That's, what I understand so far: We take any arbitrary pencil $\{Y_t \}_{t \in \mathbb{CP}^1}$ of hyperplanes in $X$ with $Y_0=Y$ (that is what we really do is we take a family $\{H_t \}_{t \in \mathbb{CP}^1}$ of hyperplanes in $\mathbb{CP}^n$ such that for every member $H_t$ the intersection $X \cap H_t$ is a $n-1$-dimensional variety with is for almost all $t$ smooth and we set $Y_t:= X \cap H_t$.
Then we consider the canonical map $\{Y_t \}_{t \in \mathbb{CP}^1} \to \mathbb{CP}^1$, remove all $t_i$ which have singular fibers (only finitely many) and make some marvelous cuts $S$ in the projective line $\mathbb{CP}^1$ making the the restriction $\{Y_t \}_{t \in \mathbb{CP}^1 ∖ S} \to \mathbb{CP}^1 ∖ S$ trivial.
Although it isn't explicitely explained I think that it is reasonable to assume that the cuts a done in that way that $\mathbb{CP}^1 ∖ S$ becomes simply connected, so the fibration becomes trivial. Is that true? Are the cuts in the original proof indeed done in same spirit like in the original construction of Riemann surfaces, to make the base simply connected?
So we consider now $\{Y_t \}_{t \in \mathbb{CP}^1 ∖ S}= Y_0 \times (\mathbb{CP}^1 ∖ S)$. Next the identifications along the slits are concretely performed. The text refers to an inductive argument. How is then the induction step done?
Next Morse theory tells us how $\{Y_t \}_{t \in \mathbb{CP}^1}$ looks like in singular points, where we take $\{Y_t \}_{t \in \mathbb{CP}^1} \to \mathbb{CP}^1$ as Morse function, right?
Well, but finally I not understand how this procedure helps to reconstruct $X$ (or say better it's homology groups) from these of $Y$ resp $\{Y_t \}_{t \in \mathbb{CP}^1}$. What was the advantage in this construction to work with Lefschetz pencil and why does it contribure a progress in the computation of the map $H_k(Y) \to H_k(X)$?