Namespaces
Variants
Actions

Difference between revisions of "Basic set"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex done)
 
(2 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 +
 
''of a linear system''
 
''of a linear system''
  
The set of points of an algebraic variety (or of a scheme) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b0153401.png" /> which belong to all the divisors of the movable part of the given linear system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b0153402.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b0153403.png" />.
+
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
 
Example. Let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b0153404.png" /></td> </tr></table>
+
$$
 +
\lambda_0 F_n(x_0, x_1, x_2) = \lambda_1 G_n(x_0, x_1, x_2) = 0
 +
$$
  
be a pencil of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b0153405.png" />-th order curves on the projective plane. The basic set of this pencil then consists of the set of common zeros of the forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b0153406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b0153407.png" />, where
+
be a pencil of $n$-th order curves on the projective plane. The basic set of this pencil then consists of the set of common zeros of the forms $F'$ and $G'$, where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b0153408.png" /></td> </tr></table>
+
$$
 +
F'. H = F_n, \qquad G'. H = G_n,
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b0153409.png" /> is the greatest common divisor of the forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b01534010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b01534011.png" />.
+
and $H$ is the greatest common divisor of the forms $F_n$ and $G_n$.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b01534012.png" /> is the rational mapping defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b01534013.png" />, then the basic set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b01534014.png" /> is the set of points of indeterminacy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b01534015.png" />. A basic set has the structure of a closed subscheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b01534016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b01534017.png" />, defined as the intersection of all divisors of the movable part of the linear system. The removal of the points of indeterminacy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b01534018.png" /> can be reduced to the trivialization of the coherent sheaf of ideals defining the subscheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b01534019.png" /> (cf. [[Birational geometry|Birational geometry]]).
+
If $\phi_L : X \to P(L)$ is the rational mapping defined by $L$, then the basic set of $L$ is the set of points of indeterminacy of $\phi_L$. A basic set has the structure of a closed subscheme $B$ in $X$, defined as the intersection of all divisors of the movable part of the linear system. The removal of the points of indeterminacy of $\phi_L$ can be reduced to the trivialization of the coherent sheaf of ideals defining the subscheme $B$ (cf. [[Birational geometry|Birational geometry]]).
  
For any linear system without fixed components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b01534020.png" /> on a smooth projective surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b01534021.png" /> there exists an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b01534022.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b01534023.png" />, then the basic set of the complete linear system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015340/b01534024.png" /> is empty (Zariski's theorem). This is not true in the multi-dimensional case.
+
For any linear system without fixed components $L$ on a smooth projective surface $F$ there exists an integer $n_0$ such that if $n > n_0$, then the basic set of the complete linear system $n L$ is empty (Zariski's theorem). This is not true in the multi-dimensional case.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> "Algebraic surfaces" ''Trudy Mat. Inst. Steklov.'' , '''75''' (1965) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Hartshorne,   "Algebraic geometry" , Springer (1977)</TD></TR></table>
+
<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>
 +
 
 +
{{TEX|done}}

Latest revision as of 04:27, 15 February 2024

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

$$ \lambda_0 F_n(x_0, x_1, x_2) = \lambda_1 G_n(x_0, x_1, x_2) = 0 $$

be a pencil of $n$-th order curves on the projective plane. The basic set of this pencil then consists of the set of common zeros of the forms $F'$ and $G'$, where

$$ F'. H = F_n, \qquad G'. H = G_n, $$

and $H$ is the greatest common divisor of the forms $F_n$ and $G_n$.

If $\phi_L : X \to P(L)$ is the rational mapping defined by $L$, then the basic set of $L$ is the set of points of indeterminacy of $\phi_L$. A basic set has the structure of a closed subscheme $B$ in $X$, defined as the intersection of all divisors of the movable part of the linear system. The removal of the points of indeterminacy of $\phi_L$ can be reduced to the trivialization of the coherent sheaf of ideals defining the subscheme $B$ (cf. Birational geometry).

For any linear system without fixed components $L$ on a smooth projective surface $F$ there exists an integer $n_0$ such that if $n > n_0$, then the basic set of the complete linear system $n L$ 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
How to Cite This Entry:
Basic set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Basic_set&oldid=12558
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article