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
Famous quotes containing the words definition, means and/or functions:
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“We have two American flags always: one for the rich and one for the poor. When the rich fly it it means that things are under control; when the poor fly it it means danger, revolution, anarchy.”
—Henry Miller (1891–1980)
“Those things which now most engage the attention of men, as politics and the daily routine, are, it is true, vital functions of human society, but should be unconsciously performed, like the corresponding functions of the physical body.”
—Henry David Thoreau (1817–1862)