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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}
Blunt cones can be excluded from the definition of convex cone by substituting "non-negative" for "positive" in the condition of α, β. A cone is called flat if it contains some nonzero vector x and its opposite −x, meaning C contains a linear subspace of dimension at least one, and salient otherwise.
In mathematics, specifically algebraic topology, the mapping cylinder [1] of a continuous function between topological spaces and is the quotient = (([,])) / where the denotes the disjoint union, and ~ is the equivalence relation generated by
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:
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.
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 ...
The normal cone C X Y or / of an embedding i: X → Y, defined by some sheaf of ideals I is defined as the relative Spec (= / +).. When the embedding i is regular the normal cone is the normal bundle, the vector bundle on X corresponding to the dual of the sheaf I/I 2.
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 ...