Algebraic Variety
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Algebraic Variety
Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ..., a sub-field of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally .... Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for ... over the real Real may re ...
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Twisted Cubic Curve
In mathematics, a twisted cubic is a smooth, rational curve ''C'' of Degree of an algebraic variety, degree three in projective space, projective 3-space P3. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation (''the'' twisted cubic, therefore). In algebraic geometry, the twisted cubic is a simple example of a projective variety that is not linear or a hypersurface, in fact not a complete intersection. It is the three-dimensional case of the rational normal curve, and is the Image (mathematics), image of a Veronese surface#Veronese map, Veronese map of degree three on the projective line. Definition The twisted cubic is most easily given parametrically as the image of the map :\nu:\mathbf^1\to\mathbf^3 which assigns to the homogeneous coordinate [S:T] the value :\nu:[S:T] \mapsto [S^3:S^2T:ST^2:T^3]. In one coordinate patch of projective space, the map is simply the moment curve :\nu:x \mapsto (x,x^2,x^3) That is, it is the cl ...
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