Divisor (algebraic Geometry) - Global Sections of Line Bundles and Linear Systems

Global Sections of Line Bundles and Linear Systems

Recall that the local equations of a Cartier divisor in a variety can be thought of as the transition maps of a line bundle, and linear equivalence as isomorphism of line bundles.

Loosely speaking, a Cartier divisor D is said to be effective if it is the zero locus of a global section of its associated line bundle . In terms of the definition above, this means that its local equations coincide with the equations of the vanishing locus of a global section.

From the divisor linear equivalence/line bundle isormorphism principle, a Cartier divisor is linearly equivalent to an effective divisor if, and only if, its associate line bundle has non-zero global sections. Two collinear non-zero global sections have the same vanishing locus, and hence the projective space over k identifies with the set of effective divisors linearly equivalent to .

If is a projective (or proper) variety over a field, then is a finite dimensional -vector space, and the associated projective space over is called the complete linear system of . Its linear subspaces are called linear systems of divisors. The Riemann-Roch theorem for algebraic curves is a fundamental identity involving the dimension of complete linear systems in the setup of projective curves.