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 | $.

## Contents

* 1 Examples.
* 1.1 References
* 1.2 Comments
* 1.3 References

### Examples.[edit]

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[edit]

[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

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[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[edit]

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[edit]

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

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