It may be checked that these three quadratic forms vanish identically when using the explicit parameterization above; that is, substitute x3 for X, and so on.
More strongly, the homogeneous ideal of the twisted cubic C is generated by these three homogeneous polynomials of degree 2.
Propertiesedit
The twisted cubic has the following properties:
It is the set-theoretic complete intersection of and , but not a scheme-theoretic or ideal-theoretic complete intersection; meaning to say that the ideal of the variety cannot be generated by only 2 polynomials; a minimum of 3 are needed. (An attempt to use only two polynomials make the resulting ideal not radical, since is in it, but is not).
Any four points on C span P3.
Given six points in P3 with no four coplanar, there is a unique twisted cubic passing through them.
The union of the tangent and secant lines (the secant variety) of a twisted cubic C fill up P3 and the lines are pairwise disjoint, except at points of the curve itself. In fact, the union of the tangent and secant lines of any non-planar smooth algebraic curve is three-dimensional. Further, any smooth algebraic variety with the property that every length four subscheme spans P3 has the property that the tangent and secant lines are pairwise disjoint, except at points of the variety itself.
The projection of C onto a plane from a point on a tangent line of C yields a cuspidal cubic.
The projection from a point on a secant line of C yields a nodal cubic.
The projection from a point on C yields a conic section.
Referencesedit
Harris, Joe (1992), Algebraic Geometry, A First Course, New York: Springer-Verlag, ISBN0-387-97716-3.
April 30, 2024
twisted, cubic, this, article, includes, list, references, related, reading, external, links, sources, remain, unclear, because, lacks, inline, citations, please, help, improve, this, article, introducing, more, precise, citations, february, 2022, learn, when,. This article includes a list of references related reading or external links but its sources remain unclear because it lacks inline citations Please help improve this article by introducing more precise citations February 2022 Learn how and when to remove this message In mathematics a twisted cubic is a smooth rational curve C of degree three in 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 of a Veronese map of degree three on the projective line Definition edit nbsp The twisted cubic is most easily given parametrically as the image of the map n P 1 P 3 displaystyle nu mathbf P 1 to mathbf P 3 nbsp which assigns to the homogeneous coordinate S T displaystyle S T nbsp the value n S T S 3 S 2 T S T 2 T 3 displaystyle nu S T mapsto S 3 S 2 T ST 2 T 3 nbsp In one coordinate patch of projective space the map is simply the moment curve n x x x 2 x 3 displaystyle nu x mapsto x x 2 x 3 nbsp That is it is the closure by a single point at infinity of the affine curve x x 2 x 3 displaystyle x x 2 x 3 nbsp The twisted cubic is a projective variety defined as the intersection of three quadrics In homogeneous coordinates X Y Z W displaystyle X Y Z W nbsp on P3 the twisted cubic is the closed subscheme defined by the vanishing of the three homogeneous polynomials F 0 X Z Y 2 displaystyle F 0 XZ Y 2 nbsp F 1 Y W Z 2 displaystyle F 1 YW Z 2 nbsp F 2 X W Y Z displaystyle F 2 XW YZ nbsp It may be checked that these three quadratic forms vanish identically when using the explicit parameterization above that is substitute x3 for X and so on More strongly the homogeneous ideal of the twisted cubic C is generated by these three homogeneous polynomials of degree 2 Properties editThe twisted cubic has the following properties It is the set theoretic complete intersection of X Z Y 2 displaystyle XZ Y 2 nbsp and Z Y W Z 2 W X W Y Z displaystyle Z YW Z 2 W XW YZ nbsp but not a scheme theoretic or ideal theoretic complete intersection meaning to say that the ideal of the variety cannot be generated by only 2 polynomials a minimum of 3 are needed An attempt to use only two polynomials make the resulting ideal not radical since Y W Z 2 2 displaystyle YW Z 2 2 nbsp is in it but Y W Z 2 displaystyle YW Z 2 nbsp is not Any four points on C span P3 Given six points in P3 with no four coplanar there is a unique twisted cubic passing through them The union of the tangent and secant lines the secant variety of a twisted cubic C fill up P3 and the lines are pairwise disjoint except at points of the curve itself In fact the union of the tangent and secant lines of any non planar smooth algebraic curve is three dimensional Further any smooth algebraic variety with the property that every length four subscheme spans P3 has the property that the tangent and secant lines are pairwise disjoint except at points of the variety itself The projection of C onto a plane from a point on a tangent line of C yields a cuspidal cubic The projection from a point on a secant line of C yields a nodal cubic The projection from a point on C yields a conic section References editHarris Joe 1992 Algebraic Geometry A First Course New York Springer Verlag ISBN 0 387 97716 3 Retrieved from https en wikipedia org w index php title Twisted cubic amp oldid 1070725748, wikipedia, wiki, book, books, library,