Linear system

A family of effective linearly equivalent divisors (cf. Divisor (algebraic geometry)) on an algebraic variety, parametrized by projective space.

Let be a non-singular algebraic variety over a field , an invertible sheaf on , the space of global sections of , and a finite-dimensional subspace. If , then the divisors determined by zero sections of are linearly equivalent and effective. A linear system is the projective space of one-dimensional subspaces of that parametrizes these divisors. If , then the linear system is said to be complete; it is denoted by .

Let be a basis of . It defines a rational mapping by the formula One usually says that is defined by the linear system . The image does not lie in any hyperplane of (see ). Conversely, every rational mapping having this property is defined by some linear system.

A fixed component of a linear system is an effective divisor on such that for any , where is an effective divisor. When runs through , the divisors form a linear system of the same dimension as . The mapping coincides with . Therefore, in considering one may assume that does not have fixed components. In this case is not defined exactly on the basic set of .

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Examples.

1) Let and , ; then the sections of can be identified with forms of degree on , and the complete linear system can be identified with the set of all curves of order .

2) The standard quadratic transformation (see Cremona transformation) is defined by the linear system of conics passing through the points , , .

3) The Geiser involution is defined by the linear system of curves of order 8 passing with multiplicity 3 through 7 points in general position (cf. Point in general position).

4) The Bertini involution is defined by the linear system of curves of order 17 passing with multiplicity 6 through 8 points in general position.