Difference between revisions of "Basic set"
From Encyclopedia of Mathematics
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''of a linear system'' | ''of a linear system'' | ||
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Example. Let | 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 | + | 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 | + | and $H$ is the greatest common divisor of the forms $F_n$ and $G_n$. |
| − | If | + | 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 | + | 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==== | ||
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</table> | </table> | ||
| − | {{TEX| | + | {{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=55504
Basic set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Basic_set&oldid=55504
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article