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Curves, dual to each other; see below for properties. In projective geometry, a dual curve of a given plane curve C is a curve in the dual projective plane consisting of the set of lines tangent to C. There is a map from a curve to its dual, sending each point to the point
These sets can be used to define a plane dual structure. Interchange the role of "points" and "lines" in C = (P, L, I) to obtain the dual structure. C ∗ = (L, P, I ∗), where I ∗ is the converse relation of I. C ∗ is also a projective plane, called the dual plane of C. If C and C ∗ are isomorphic, then C is called self-dual.
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A.
A curve in this context is defined by a non-degenerate algebraic equation in the complex projective plane. Lines in this plane correspond to points in the dual projective plane and the lines tangent to a given algebraic curve C correspond to points in an algebraic curve C * called the dual curve.
In Euclidean and projective geometry, five points determine a conic (a degree-2 plane curve), just as two (distinct) points determine a line (a degree-1 plane curve).There are additional subtleties for conics that do not exist for lines, and thus the statement and its proof for conics are both more technical than for lines.
The n-torsion of an abelian variety and the n-torsion of its dual are dual to each other when n is coprime to the characteristic of the base. In general — for all n — the n-torsion group schemes of dual abelian varieties are Cartier duals of each other. This generalises the Weil pairing for elliptic curves.