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In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme X , the relative Spec C = Spec X R {\displaystyle C=\operatorname {Spec} _{X}R}
From the elementary properties of convex cones, C is the interior of its closure and is a proper cone. The elements in the closure of C are precisely the square of elements in E. C is self-dual. In fact the elements of the closure of C are just set of all squares x 2 in E, the dual cone is given by all a such that (a,x 2) > 0.
The cone over a closed interval I of the real line is a filled-in triangle (with one of the edges being I), otherwise known as a 2-simplex (see the final example). The cone over a polygon P is a pyramid with base P. The cone over a disk is the solid cone of classical geometry (hence the concept's name). The cone over a circle given by
An affine convex cone is the set resulting from applying an affine transformation to a convex cone. [8] A common example is translating a convex cone by a point p: p + C. Technically, such transformations can produce non-cones. For example, unless p = 0, p + C is not a linear cone. However, it is still called an affine convex cone.
A chemical element, often simply called an element, is a type of atom which has a specific number of protons in its atomic nucleus (i.e., a specific atomic number, or Z). [ 1 ] The definitive visualisation of all 118 elements is the periodic table of the elements , whose history along the principles of the periodic law was one of the founding ...
This can be much easier: for example, if X is regularly embedded in Y then its normal cone is a vector bundle, so we are reduced to the problem of finding the intersection product of a subscheme C W V of a vector bundle C X Y with the zero section X. However this intersection product is just given by applying the Gysin isomorphism to C W V.
A cone C in a vector space X is said to be self-dual if X can be equipped with an inner product ⋅,⋅ such that the internal dual cone relative to this inner product is equal to C. [3] Those authors who define the dual cone as the internal dual cone in a real Hilbert space usually say that a cone is self-dual if it is equal to its internal dual.
Consider, for example, the case where Y is the disk , and : = is the standard inclusion of the circle as the boundary of . Then the mapping cone C f {\displaystyle C_{f}} is homeomorphic to two disks joined on their boundary, which is topologically the sphere S 2 {\displaystyle S^{2}} .