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I'm reading an article of mumford and I want to know what it means or where I can find: the pencil has no fixed components, and the pencil is fixed components.


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Hi, Nicolas and welcome to MO. I would recommend adding a more exact reference to the article, there are several Mumfords who have published in mathematics and they all have several published articles. – Rami Luisto Feb 20 '13 at 10:38
@Rami: I guess that "mumford" should be the most unsignificant of all Mumfords. – Sasha Feb 20 '13 at 11:20
the article is Further pathologies in algebraic geometry, is the second exemple about the chacarteristic map. thanks – Nicolas Feb 20 '13 at 12:28

Assuming that you are familiar with the notion of linear system of divisors, a pencil is a linear system of (projective) dimension $1$. One says that a pencil (or an arbitrary linear system) on a variety $X$ has a fixed component $D$, where $D$ is, say, an effective Cartier divisor (if $X$ is smooth, you may repace "effective Cartier divisor" by "irreducible subvariety of codimension one") if all the divisors of the linear system in question contain $D$.

The basic definitions concerning linear systems may be found in almost any textbook (Hartshorne will do).

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Thanks Serge, Yes, I am familiar with the notion of linear system and cartier divisor. – Nicolas Feb 20 '13 at 12:54

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