Difference between revisions of "Basic set"
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| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> "Algebraic surfaces" ''Trudy Mat. Inst. Steklov.'' , '''75''' (1965) (In Russian) {{MR|}} {{ZBL|0154.33002}} {{ZBL|0154.21001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table> |
Revision as of 21:50, 30 March 2012
of a linear system
The set of points of an algebraic variety (or of a scheme) 
which belong to all the divisors of the movable part of the given linear system 
on 
.
Example. Let
 |
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
 |
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