Divisors, line bundles, and linear systems

An effective Cartier divisor is a closed subscheme whose ideal sheaf is invertible. Locally it is defined by one nonzerodivisor f, so multiplication by f embeds the local ring into itself. Being locally principal is insufficient: a zero divisor can vanish along an entire irreducible component and does not give the required invertible ideal. The condition concerns the ambient local ring, not whether a derivative of f vanishes or whether a drawing looks like a hypersurface.

A general Cartier divisor is specified by invertible meromorphic local equations whose ratios on overlaps are regular units. Its associated invertible sheaf O(D) glues using these ratios. For an effective divisor there is a regular section of O(D) whose zero scheme recovers it, and the ideal sheaf is O(−D). The zero scheme can be nonreduced: a square of a parameter in a domain is a nonzerodivisor even though its vanishing has multiplicity two.

On a Noetherian integral scheme a Weil divisor is a finite integer combination of codimension-one integral closed subschemes. If the scheme is only locally Noetherian, locally finite combinations are needed instead. On a normal such scheme the local rings at codimension-one points are discrete valuation rings, giving orders of zeros and poles of rational functions. Normality does not imply that every Weil divisor is Cartier: a codimension-one combination can fail to admit compatible local principal equations near singular points.

On a smooth integral curve over an algebraically closed field, the local rings at closed points are discrete valuation rings and the two divisor notions agree. A divisor is a finite sum of points, and its degree is the sum of coefficients in this field setting. Over a non-algebraically-closed field, closed-point residue degrees must weight that sum. Principal divisors have degree zero on a smooth projective curve. Linear equivalence quotients by such principal divisors and identifies divisors that give isomorphic associated line bundles.

Choose a finite-dimensional vector space V of global sections of a line bundle L. A point is a base point when every section in V vanishes in its fiber. If V generates L everywhere, the sections define a morphism into a projective space, with the usual choice of dual convention. Without that generation property they define only a rational map on the open where not all values vanish. Counting sections alone does not prove that the map separates points, separates tangent directions, or is an embedding.

For the projective line, the sections of O(2) are homogeneous quadratics. A basis gives the familiar conic map [s:t]↦[s²:st:t²], illustrating the relation between a line bundle and a projective presentation. Choosing a smaller subspace can introduce base points or identify points, even though the underlying line bundle has not changed. Consequently the divisor class, the line bundle, the chosen space of sections, and the resulting morphism are related objects with distinct information.