nLab pencil

Rough idea

A pencil is a 1-parametric family of divisors in algebraic geometry.

Definition

A pencil is a linear system of divisors of dimension $1$, that is parametrized by a projective line.

Examples

Pencil of lines through a given point

The basic example is the pencil of all lines passing through a single point in a projective plane, or in an affine plane.

The pencil of lines which pass through a point at infinity in projective plane consists from the point of view of an affine complement of the line at infinity to an equivalence class of parallel lines.

Conversely the projective completion of any affine space is obtained by adding points at infinity. Each point can be identified with the set of all lines through it (this is related to the projective duality). A point at infinity attached to an affine space is defined as the equivalence class of parallel lines (see direction of a line) which may be viewed as those lines in the affine space which are pointing toward that point at infinity.

Pencils of conics

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