Linear system

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

Let $ X $ be a non-singular algebraic variety over a field $ k $, $ {\mathcal L} $ an invertible sheaf on $ X $, $ \Gamma ( X , {\mathcal L} ) $ the space of global sections of $ {\mathcal L} $, and $ L \subset \Gamma ( X , {\mathcal L} ) $ a finite-dimensional subspace. If $ \mathop{\rm dim} L > 0 $, then the divisors determined by zero sections of $ L $ are linearly equivalent and effective. A linear system is the projective space $ | L | = P ( L) $ of one-dimensional subspaces of $ L $ that parametrizes these divisors. If $ \mathop{\rm dim} \Gamma ( X , {\mathcal L} ) < \infty $, then the linear system $ | \Gamma ( X , {\mathcal L} ) | $ is said to be complete; it is denoted by $ | L | $.

Let $ s _ {0} \dots s _ {n} $ be a basis of $ L $. It defines a rational mapping $ \phi _ {L} : X \rightarrow P ^ {n} $ by the formula

$$ x \mapsto ( s _ {0} ( x) \dots s _ {n} ( x) ) ,\ \ x \in X . $$

One usually says that $ \phi _ {L} $ is defined by the linear system $ | L | $. The image $ \phi _ {L} ( X) $ does not lie in any hyperplane of $ P ^ {n} $( see [2]). Conversely, every rational mapping $ \psi : X \rightarrow P ^ {m} $ having this property is defined by some linear system.

A fixed component of a linear system $ | L | $ is an effective divisor $ D ^ {*} $ on $ X $ such that $ D = D ^ \prime + D ^ {*} $ for any $ D \in | L | $, where $ D ^ \prime $ is an effective divisor. When $ D $ runs through $ | L | $, the divisors $ D ^ \prime $ form a linear system $ | L ^ \prime | $ of the same dimension as $ | L | $. The mapping $ \phi _ {L ^ \prime } $ coincides with $ \phi _ {L} $. Therefore, in considering $ \phi _ {L} $ one may assume that $ | L | $ does not have fixed components. In this case $ \phi _ {L} $ is not defined exactly on the basic set of $ | L | $.

Examples.

1) Let $ X = P ^ {2} $ and $ L = {\mathcal O} _ {P ^ {2} } ( d) $, $ d \geq 1 $; then the sections of $ \Gamma ( P ^ {2} , {\mathcal O} _ {P ^ {2} } ( d) ) $ can be identified with forms of degree $ d $ on $ P ^ {2} $, and the complete linear system $ ( {\mathcal O} _ {P ^ {2} } ( d) ) $ can be identified with the set of all curves of order $ d $.

2) The standard quadratic transformation $ \tau : P ^ {2} \rightarrow P ^ {2} $( see Cremona transformation) is defined by the linear system of conics passing through the points $ ( 0 , 0 , 1 ) $, $ ( 0 , 1 , 0 ) $, $ ( 1 , 0 , 0 ) $.

3) The Geiser involution $ \alpha : P ^ {2} \rightarrow P ^ {2} $ 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 $ \beta : P ^ {2} \rightarrow P ^ {2} $ is defined by the linear system of curves of order 17 passing with multiplicity 6 through 8 points in general position.

References

[1]	I.R. Shafarevich, "Algebraic surfaces" Proc. Steklov Inst. Math. , 75 (1967) Trudy Mat. Inst. Steklov. , 75 (1965) MR1392959 MR1060325 Zbl 0830.00008 Zbl 0733.14015 Zbl 0832.14026 Zbl 0509.14036 Zbl 0492.14024 Zbl 0379.14006 Zbl 0253.14006 Zbl 0154.21001
[2]	D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) MR0209285 Zbl 0187.42701
[3]	O. Zariski, "Algebraic surfaces" , Springer (1971) MR0469915 Zbl 0219.14020

Comments

In classical (elementary) projective and analytic geometry one speaks of linear systems of curves, surfaces, quadrics, etc. These are families of curves, surfaces, etc. of the form

$$ \lambda _ {1} F _ {1} + \dots + \lambda _ {m} F _ {m} = 0 , $$

where the $ F _ {i} = 0 $ define individual curves, surfaces, etc. If the family is one-dimensional (i.e. through a point in general position passes one member of the family), one speaks of a pencil; a two-dimensional family (i.e. two different members of the family pass through a point in general position) is called a net; and a three- (or higher-) dimensional family is called a web, [a1]. Instead of "net" the term "bundlebundle" is also occasionally used and instead of "web" one also sometimes finds "net" .

Quite generally, if $ U $ is an open subset of $ \mathbf R ^ {n} $, a codimension $ k $ $ d $- web on $ U $ is defined by $ d $ foliations of codimension $ k $ on $ U $ such that for each $ x \in U $ the $ d $ leaves passing through $ x $ are in general position. Cf. also Web. Especially in the case of a codimension $ ( n - 1) $ $ n $- web, i.e. an $ n $- web of curves, on $ U \subset \mathbf R ^ {n} $( same $ n $) the word net is often used.

The phrase "linear system" of course also occurs (as an abbreviation) in many other parts of mathematics. E.g. in differential equation theory: for system of linear differential equations, and in control and systems theory: for linear input/output systems, linear dynamical systems or linear control system.

References

[a1]	J.A. Todd, "Projective and analytical geometry" , Pitman (1947) pp. Chapt. VI MR1527119 MR0024624 Zbl 0031.06701