Definition
More precisely, suppose that is a linear system of divisors on some variety . Consider the intersection
where denotes the support of a divisor, and the intersection is taken over all effective divisors in the linear system. This is the base locus of (as a set, at least: there may be more subtle scheme-theoretic considerations as to what the structure sheaf of should be).
One application of the notion of base locus is to nefness of a Cartier divisor class (i.e. complete linear system). Suppose is such a class on a variety, and an irreducible curve on . If is not contained in the base locus of, then there exists some divisor in the class which does not contain, and so intersects it properly. Basic facts from intersection theory then tell us that we must have . 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 of (Cartier) divisors on a variety is viewed as a line bundle on . From this viewpoint, the base locus is the set of common zeroes of all sections of . A simple consequence is that the bundle is globally generated if and only if the base locus is empty.
Read more about this topic: Base Locus
Famous quotes containing the word definition:
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)