RATIONAL NORMAL CURVE Meaning and
Definition
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A rational normal curve is a concept in algebraic geometry that refers to a particular type of algebraic curve in projective space. In simple terms, it is a smooth, non-degenerate curve that lies entirely within a projective space of a given dimension.
More precisely, a rational normal curve is defined as a projective curve parameterized by rational functions, where the coordinates of the curve are expressed by ratios of polynomials. These rational functions are chosen such that the curve possesses certain desirable properties, including being non-singular and lying entirely within a given projective space.
The term "normal" in rational normal curve indicates that it is non-singular, meaning that it has no singular points or self-intersections. Additionally, it is "rational" because the curve is parameterized by rational functions, which can be expressed as ratios of polynomials.
Rational normal curves have been extensively studied in algebraic geometry and are particularly useful in various mathematical applications. They arise naturally in several contexts, including in the study of linear systems of divisors on algebraic curves and in the construction of higher-dimensional varieties. They also play an important role in the theory of secant varieties, which are algebraic varieties associated with the rational normal curve. Due to their special properties, rational normal curves are often used as foundational objects for studying and understanding more general curves and varieties in algebraic geometry.
