the line bundle on X associated to a hyperplane section (the intersection of X with a hyperplane in ). Basepoint-free line bundles Let X be a scheme over a field k (for example, an algebraic variety) with a line bundle L. (A line bundle may also be called an invertible sheaf.) Let be elements of the k-vector space of global sections of L. The zero set of each section is a closed subset of X; let U be the open subset of points at which at least one of is not zero. Then these sections define a morphism In more detail: for each point x of U, the fiber of L over x is a 1-dimensional vector space over the residue field k(x). Choosing a basis for this fiber makes into a sequence of n+1 numbers, not all zero, and hence a point in projective space. Changing the choice of basis scales all the numbers by the same nonzero constant, and so the point in projective space is independent of the choice. Moreover, this morphism has the property that the restriction of L to U is isomorphic to the pullback . The base locus of a line bundle L on a scheme X is the intersection of the zero sets of all global sections of L. A line bundle L is called basepoint-free if its base locus is empty. That is, for every point x of X there is a global section of L which is nonzero at x. If X is proper over a field k, then the vector space of global sections has finite dimension; the dimension is called . So a basepoint-free line bundle L determines a morphism over k, where , given by choosing a basis for . Without making a choice, this can be described as the morphism from X to the space of hyperplanes in , canonically associated to the basepoint-free line bundle L. This morphism has the property that L is the pullback . Conversely, for any morphism f from a scheme X to projective space over k, the pullback line bundle is basepoint-free. Indeed, O(1) is basepoint-free on , because for every point y in there is a hyperplane not containing y. Therefore, for every point x in X, there is a section s of O(1) over that is not zero at f(x), and the pullback of s is a global section