Basepoint

Basepoint, in mathematics, is a distinguished point chosen in a space or family of objects to serve as a reference. In topology and algebraic topology, a basepoint is a point x0 in a topological space X, used to form pointed spaces (X, x0) and to define based maps, loops, and invariants such as the fundamental group π1(X, x0). The choice of basepoint matters in spaces that are not path-connected; for path-connected spaces, different basepoints yield isomorphic fundamental groups, with isomorphisms determined by a path between the basepoints.

In this setting, the based loop space Ω(X, x0) consists of loops starting and ending at x0, and concatenation of based loops gives a group-like structure that underpins homotopy theory.

In algebraic geometry, a basepoint (or base locus) of a linear system on a variety is a common zero of all sections in the family. If every divisor in a linear system passes through a fixed point, that point is a basepoint. Basepoints can prevent a morphism from being defined everywhere, and they are often removed by a blow-up, introducing an exceptional divisor and yielding a new space where the linear system becomes basepoint-free.

In projective geometry, pencils, nets, and other linear systems may have basepoints where all members meet; studying the base locus informs the geometry of the associated map to projective space.

In broader usage, a basepoint can simply be a reference point used to anchor constructions or measurements, though the specific meaning depends on context.