In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family. These arose first in the form of a linear system of algebraic curves in the projective plane. It assumed a more general form, through gradual generalisation, so that one could speak of linear equivalence of divisors D on a general scheme or even a ringed space ( X , O X ) {\displaystyle (X,{\mathcal {O}}_{X})} . Linear systems of dimension 1, 2, or 3 are called a pencil, a net, or a web, respectively. A map determined by a linear system is sometimes called the Kodaira map.
Definitions Given a general variety X {\displaystyle X} , two divisors D , E ∈ Div ( X ) {\displaystyle D,E\in {\text{Div}}(X)} are linearly equivalent if
E = D + ( f ) {\displaystyle E=D+(f)\ }
for some non-zero rational function f {\displaystyle f} on X {\displaystyle X} , or in other words a non-zero element f {\displaystyle f} of the function field k ( X ) {\displaystyle k(X)} . Here ( f ) {\displaystyle (f)} denotes the divisor of zeroes and poles of the function f {\displaystyle f} . Note that if X {\displaystyle X} has singular points, the notion of '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 X {\displaystyle X} is defined as the set of all effective divisors linearly equivalent to some given divisor D ∈ Div ( X ) {\displaystyle D\in {\text{Div}}(X)} . It is denoted | D | {\displaystyle |D|} . Let L {\displaystyle {\mathcal {L}}} be the line bundle associated to D {\displaystyle D} . In the case that X {\displaystyle X} is a nonsingular projective variety, the set | D | {\displaystyle |D|} is in natural bijection with ( Γ ( X , L ) ∖ { 0 } ) / k ∗ , {\displaystyle (\Gamma (X,{\mathcal {L}})\smallsetminus \{0\})/k^{\ast },} by associating the element E = D + ( f ) {\displaystyle E=D+(f)} of | D | {\displaystyle |D|} to the set of non-zero multiples of f {\displaystyle f} (this is well defined since two non-zero rational functions have the same divisor if and only if they are non-zero multiples of each other). A complete linear system | D | {\displaystyle |D|} is therefore a projective space. A linear system d {\displaystyle {\mathfrak {d}}} is then a projective subspace of a complete linear system, so it corresponds to a vector subspace W of Γ ( X , L ) . {\displaystyle \Gamma (X,{\mathcal {L}}).} The dimension of the linear system d {\displaystyle {\mathfrak {d}}} is its dimension as a projective space. Hence dim d = dim W − 1 {\displaystyle \dim {\mathfrak {d}}=\dim W-1} . Linear systems can also be introduced by means of the line bundle or invertible sheaf language. In those terms, divisors D {\displaystyle D} (Cartier divisors, to be precise) correspond to line bundles, and linear equivalence of two divisors means that the corresponding line bundles are isomorphic.
Examples
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