by the vanishing locus of some using the rational function (Proposition 7.2). For example, the divisor associated to the vanishing locus of is linearly equivalent to the divisor associated to the vanishing locus of . Then, there is the equivalence of divisors Linear systems on curves One of the important complete linear systems on an algebraic curve of genus is given by the complete linear system associated with the canonical divisor , denoted . This definition follows from proposition II.7.7 of Hartshorne since every effective divisor in the linear system comes from the zeros of some section of . Hyperelliptic curves One application of linear systems is used in the classification of algebraic curves. A hyperelliptic curve is a curve with a degree morphism . For the case all curves are hyperelliptic: the Riemann–Roch theorem then gives the degree of is and , hence there is a degree map to . grd A is a linear system on a curve which is of degree and dimension . For example, hyperelliptic curves have a since defines one. In fact, hyperelliptic curves have a unique from proposition 5.3. Another close set of examples are curves with a which are called trigonal curves. In fact, any curve has a for . Linear systems of hypersurfaces in a projective space Consider the line bundle over . If we take global sections , then we can take its projectivization . This is isomorphic to where Then, using any embedding we can construct a linear system of dimension . Linear system of conics Characteristic linear system of a family of curves The characteristic linear system of a family of curves on an algebraic surface Y for a curve C in the family is a linear system formed by the curves in the family that are infinitely near C. In modern terms, it is a subsystem of the linear system associated to the normal bundle to . Note a characteristic system need not to be complete; in fact, the question of completeness is something studied extensively by the Italian school without a satisfactory conclusion; nowadays, the Kodaira–Spencer theory can be used to answer the question of the completeness. Other examples The Cayley–Bacharach theorem is a property of a pencil of cubics, which states that the base locus satisfies an "8 implies 9" property: any cubic containing 8 of the points necessarily contains the 9th. Linear systems in