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Let $X \subset \mathbb{P}^n$ be projective variety over alg closed field of char $0$ and $C = V(F), D= V(G) \subset X$ two distinct divisors (e.g. two quadrics, curves or lines lying in a surface,...) where $F,G \in \mathbb{C}[X_0,...,X_n]$ are homogeneous polynomials.

Then we can construct in natural way the pencil of $C,D$ as subvariety

$$V( \alpha F+\beta G) \subset \mathbb{P}^1 \times X $$

with $\alpha, \beta \in \mathbb{C}$. What are the main motivations to introduce this new algebraic set? Or say are there any reasons to introduce this new object in order to obtain fruitful results and insights about structure of divisors $C, D$ and $X$ which are presumably much harder to reach without the pencil concept?

The question is identical to that one got little attention.

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    $\begingroup$ It would be nice if we could define an equivalence relation $C\equiv D$ if they are related via a pencil as you've described. This isn't quite an equivalence relation, so let's define $C\equiv D$ if there's a chain of relations as you've described linking $C$ to $D_1$, then $D_1$ to $D_2$, etc, until you end up at $D$. That will be an equivalence relation, it's called rational equivalence. It's not hard to see that linearly equivalent divisors will be $\equiv$ equivalent. So you can form the quotient group $\operatorname{Pic}(X)/\equiv$. This group has been much studied. $\endgroup$
    – Joe Silverman
    Commented Feb 14, 2021 at 0:43
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    $\begingroup$ Perhaps Lefschetz pencils are an example of something pencils are useful for. $\endgroup$
    – Zach Teitler
    Commented Feb 14, 2021 at 1:13

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