Pencil (mathematics)

A pencil in projective geometry is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a projective plane.

More generally, a pencil is the special case of a linear system of divisors in which the parameter space is a projective line. Typical pencils of curves in the projective plane, for example, are written as

where

are plane curves.

A pencil of planes, the family of planes through a given straight line, is sometimes referred to as a fan or a sheaf.