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

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

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