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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
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}
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.
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:
As the cone is by definition Kähler, there exists a complex structure J. The Reeb vector field on the Sasaskian manifold is defined to be = (/). It is nowhere vanishing. It commutes with all holomorphic Killing vectors on the cone and in particular with all isometries of the Sasakian manifold. If the orbits of the vector field close then the ...
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.
The cone of curves is defined to be the convex cone of linear combinations of curves with nonnegative real coefficients in the real vector space () of 1-cycles modulo numerical equivalence. The vector spaces N 1 ( X ) {\displaystyle N^{1}(X)} and N 1 ( X ) {\displaystyle N_{1}(X)} are dual to each other by the intersection pairing, and the nef ...
A more involved example is the role played by the cone of curves in the theory of minimal models of algebraic varieties. Briefly, the goal of that theory is as follows: given a (mildly singular) projective variety X {\displaystyle X} , find a (mildly singular) variety X ′ {\displaystyle X'} which is birational to X {\displaystyle X} , and ...
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