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.
Thanks
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. Thanks 


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). 

