# What's the meaning of pencils in birational geometry?

I see in some books the authors call a one dimensional linear system a pencil, but in other books one call a linear system $|D|$ is not compsited with a pencil if $\dim \phi_{|D|}(X)\geq 2$ and even someone just say a pencil of curves etc. It seems these terms have different meanings. My question is the following.

What is the common pointview about the notion of a pencil? What is the realtions among above terms?

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Well, I think this is kind of an antiquated terminology and I think it is going out of use (hopefully). I believe the classical meaning of a pencil was a 1-dimensional linear system of curves (whose members are connected) on a surface. In other words, calling the equivalent notion a pencil (in arbitrary dimensions) is just a generalization of the notion of a pencil of curves. The only not-necessarily-obvious usage is the expression not-composite-with-a-pencil. The point is, that by a pencil of curves they meant a surface fibred over a curve with connected fibers. I am sure that when this was originally invented they did not yet talk about fibrations and such. Anyway, the point is, if you say your linear system is one dimensional it could still be a fibration followed by a finite map, that is, the composition of a pencil and another map, so your original linear system is composite-with-a-pencil.

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S&aacute;ndor: I recommend clarifying the phrase "... which is incidentally a $1$-dimensional linear system ..." Since it is not true that every one-dimensional family of curves on a surface is a $1$-dimensional linear system, I am worried that some readers could misunderstand you. –  Jason Starr Sep 8 '12 at 14:34
@Jason: thanks for the suggestion. I edited the text. Probably still not ideal, but perhaps less misleading.... –  Sándor Kovács Sep 8 '12 at 17:33
There are people still talking that's divisor composed of a pencil. I have some confusion with that. –  MZWang Sep 10 '12 at 12:11
Dear Sandor, thanks for your clarification. –  MZWang Sep 10 '12 at 12:16
@MZWang: I suppose a "divisor composed of a pencil" could mean that it's complete linear system is such, although this sounds like some abuse of language. –  Sándor Kovács Sep 10 '12 at 14:41

There is a difference between the notion of "linear pencil" and "algebraic pencil." Roughly, a linear pencil on $X$ is just a linear series of dimension 1; equivalently, it can be thought of as a map $X\to \mathbb P^1$. On the other hand, an algebraic pencil is either a map $X\to C$ for a curve $C$, or equivalently, the family of fibers of such a map.

The terminology "not composite with a pencil" refers to the algebraic kind of pencil, not the linear kind.

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A remark on the last sentence: "not composite with a pencil" is use also for linear pencils. –  rita Sep 8 '12 at 17:34
Is it ever used specifically for linear pencils, or are you just remarking that a linear pencil is a special case of an algebraic pencil? –  Jack Huizenga Sep 8 '12 at 19:35
@Jack: it is often used without distinguishing the linear case (including case when there are base points) from the general one. Sometimes people talk about "rational" (i.e. over ${\mathbb P^1$) and "irrational" pencils (over a curve of genus $>0$). –  rita Sep 9 '12 at 10:23
@Jack: I prefer this explanation. Although I still have some confusion about a divisor composed of a pencil. –  MZWang Sep 10 '12 at 12:15