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A cone is a convex cone if + belongs to , for any positive scalars , , and any , in . [5] [6] A cone is convex if and only if +.This concept is meaningful for any vector space that allows the concept of "positive" scalar, such as spaces over the rational, algebraic, or (more commonly) the real numbers.
In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme X, ... Then, by definition, the projective cone of R is: () ...
The definition of a cone may be extended to higher dimensions; see convex cone. In this case, one says that a convex set C in the real vector space R n {\displaystyle \mathbb {R} ^{n}} is a cone (with apex at the origin) if for every vector x in C and every nonnegative real number a , the vector ax is in C . [ 2 ]
The definition of the tangent cone can be extended to abstract algebraic varieties, and even to general Noetherian schemes. Let X be an algebraic variety, x a point of X, and (O X,x, m) be the local ring of X at x. Then the tangent cone to X at x is the spectrum of the associated graded ring of O X,x with respect to the m-adic filtration:
The vector spaces () and () are dual to each other by the intersection pairing, and the nef cone is (by definition) the dual cone of the cone of curves. [ 6 ] A significant problem in algebraic geometry is to analyze which line bundles are ample , since that amounts to describing the different ways a variety can be embedded into projective space.
In algebraic geometry, the normal cone of a subscheme of a scheme is a scheme analogous to the normal bundle or tubular neighborhood in differential geometry.
Ruled surfaces appear in the Enriques classification of projective complex surfaces, because every algebraic surface of Kodaira dimension is a ruled surface (or a projective plane, if one uses the restrictive definition of ruled surface). Every minimal projective ruled surface other than the projective plane is the projective bundle of a 2 ...
Cone of a circle. The original space X is in blue, and the collapsed end point v is in green.. In topology, especially algebraic topology, the cone of a topological space is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to a point.
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