Basepoints

Basepoints, in algebraic geometry, refer to the points common to every member of a linear system on a variety. If X is a projective variety over an algebraically closed field and L is a linear system (a finite-dimensional subspace of sections of a line bundle, often written |D| for a divisor D), the base locus B(L) is the set of points x in X where every divisor in L passes through x. Equivalently, B(L) is the closed subscheme defined by the base ideal I_L, the image of the evaluation map H^0(X, L) ⊗ O_X → L.

The base locus captures where the linear system fails to move. A linear system is basepoint free if B(L) is empty; in that case L defines a morphism from X to projective space, given by the global sections of L. If B(L) is nonempty, the associated map is only a rational map, undefined at B(L). Such indeterminacies can often be resolved by blowing up X along the base locus, after which the pulled-back system may become basepoint free.

A standard way to view a linear system is as a fixed part plus a movable part. One writes |D| = F + |M|, where F is the fixed (base) part and |M| is movable. The base locus is supported by the fixed part, and removing fixed components of the system can render the remaining system basepoint free.

Basepoints play a central role in constructing embeddings and morphisms. A very ample divisor has a basepoint free complete linear system that embeds X into projective space; more generally, basepoint freeness is a criterion for when a divisor yields a morphism rather than a rational map. The concept is also essential in birational geometry and the resolution of singularities, where blowing up along base loci is a common technique.