Basic set
of a linear system
The set of points of an algebraic variety (or of a scheme) $X$ which belong to all the divisors of the movable part of the given linear system $L$ on $X$.
Example. Let
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be a pencil of -th order curves on the projective plane. The basic set of this pencil then consists of the set of common zeros of the forms
and
, where
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and is the greatest common divisor of the forms
and
.
If is the rational mapping defined by
, then the basic set of
is the set of points of indeterminacy of
. A basic set has the structure of a closed subscheme
in
, defined as the intersection of all divisors of the movable part of the linear system. The removal of the points of indeterminacy of
can be reduced to the trivialization of the coherent sheaf of ideals defining the subscheme
(cf. Birational geometry).
For any linear system without fixed components on a smooth projective surface
there exists an integer
such that if
, then the basic set of the complete linear system
is empty (Zariski's theorem). This is not true in the multi-dimensional case.
References
[1] | "Algebraic surfaces" Trudy Mat. Inst. Steklov. , 75 (1965) (In Russian) Zbl 0154.33002 Zbl 0154.21001 |
[2] | R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001 |
Basic set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Basic_set&oldid=23761