Definition By Means of Functions
Given the fundamental idea of a rational function on a general variety V, or in other words of a function f in the function field of V, divisors D and E are linearly equivalent if
where (f) denotes the divisor of zeroes and poles of the function f.
Note that if V has singular points, 'divisor' is inherently ambiguous (Cartier divisors, Weil divisors: see divisor (algebraic geometry)). The definition in that case is usually said with greater care (using invertible sheaves or holomorphic line bundles); see below.
A complete linear system on V is defined as the set of all effective divisors linearly equivalent to some given divisor D. It is denoted |D|. Let L(D) be the line bundle associated to D. It can be proved that |D| corresponds bijectively to and is therefore a projective space.
A linear system is then a projective subspace of a complete linear system, so it corresponds to a vector subspace W of The dimension of the linear system is its dimension as a projective space. Hence .
Read more about this topic: Linear System Of Divisors
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