Linear system
A family of effective linearly equivalent divisors (cf. Divisor) 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 [2]). 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
.
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.
References
[1] | I.R. Shafarevich, "Algebraic surfaces" Proc. Steklov Inst. Math. , 75 (1967) Trudy Mat. Inst. Steklov. , 75 (1965) |
[2] | D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) |
[3] | O. Zariski, "Algebraic surfaces" , Springer (1971) |
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
![]() |
where the 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 is an open subset of
, a codimension
-web on
is defined by
foliations of codimension
on
such that for each
the
leaves passing through
are in general position. Cf. also Web. Especially in the case of a codimension
-web, i.e. an
-web of curves, on
(same
) 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 |
Linear system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_system&oldid=24498