Base locus

In mathematics, specifically algebraic geometry, the base locus of a linear system of divisors on a variety refers to the subvariety of points 'common' to all divisors in the linear system.

Geometrically, this corresponds to the common intersection of the varieties.

Definition

More precisely, suppose that $[D]$ is a linear system of divisors on some variety $X$. Consider the intersection

$\textrm{Bl} \ ([D]) := \bigcap_{D_\text{eff} \in [D]} \textrm{ Supp } \ D_\text{eff} \$

where $\textrm{Supp}$ denotes the support of a divisor, and the intersection is taken over all effective divisors $D_\text{eff}$ in the linear system. This is the base locus of $[D]$ (as a set, at least: there may be more subtle scheme-theoretic considerations as to what the structure sheaf of $\textrm{Bl}$ should be).

One application of the notion of base locus is to nefness of a Cartier divisor class (i.e. complete linear system). Suppose $[D]$ is such a class on a variety $X$, and $C$ an irreducible curve on $X$. If $C$ is not contained in the base locus of $[D]$, then there exists some divisor $\tilde D$ in the class which does not contain $C$, and so intersects it properly. Basic facts from intersection theory then tell us that we must have $[D] \cdot C \geq 0$. The conclusion is that to check nefness of a divisor class, it suffices to compute the intersection number with curves contained in the base locus of the class. So, roughly speaking, the 'smaller' the base locus, the 'more likely' it is that the class is nef.

In the modern formulation of algebraic geometry, a linear system $[D]$ of (Cartier) divisors on a variety $X$ is viewed as a line bundle $L_{[D]}$on $X$. From this viewpoint, the base locus $\textrm{Bl} \ ([D])$ is the set of common zeroes of all sections of $L_{[D]}$. A simple consequence is that the bundle is globally generated if and only if the base locus is empty.

References

• Hartshorne, R. Algebraic Geometry, Springer-Verlag, 1977; corrected 6th printing, 1993. ISBN 0-387-90244-9.
• Lazarsfeld, R., Positivity in Algebraic Geometry I, Springer-Verlag, 2004. ISBN 3-540-22533-1.