# 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 ). 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 $| L |$.

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### 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.

How to Cite This Entry:
Linear system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_system&oldid=47664
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article