View source for (heparan sulfate)-glucosamine 3-sulfotransferase 1

{{mergeto|Linear system of divisors|date=February 2013}}
{{main|Linear system of divisors}}
In [[mathematics]], specifically [[algebraic geometry]], the '''base locus''' of a [[linear system of divisors]] on a [[Algebraic variety|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 <math>[D]</math> is a linear system of divisors on some variety <math>X</math>. Consider the intersection 

: <math>\textrm{Bl} \ ([D]) := \bigcap_{D_\text{eff} \in [D]} \textrm{ Supp } \ D_\text{eff} \ </math>

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

One application of the notion of base locus is to [[Numerically effective|nefness]] of a Cartier divisor class (i.e. complete linear system). Suppose <math>[D]</math> is such a class on a variety <math>X</math>, and <math>C</math> an irreducible curve on <math>X</math>. If <math>C</math> is not contained in the base locus of <math>[D]</math>, then there exists some divisor <math>\tilde D</math> in the class which does not contain <math>C</math>, and so intersects it properly. Basic facts from intersection theory then tell us that we must have <math>[D] \cdot C \geq 0</math>. 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 <math>[D]</math> of (Cartier) divisors on a variety <math>X</math> is viewed as a line bundle <math>L_{[D]}</math>on <math>X</math>. From this viewpoint, the base locus <math>\textrm{Bl} \ ([D])</math> is the set of common zeroes of all sections of <math>L_{[D]}</math>. A simple consequence is that the bundle is [[Ample line bundle|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.

[[Category:Algebraic geometry]]