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Let be a field. A conic section over is the zero set of a degree 2 polynomial in the affine plane , or better yet the zero set of a homogeneous polynomial of degree 2 in the projective plane .
In the classical case of real numbers, conic sections may be pictured in terms of intersections of a standard cone in affine 3-space with various affine hyperplanes (hence the name, “conic section”). In this picture, nonsingular conic sections are classified (up to automorphisms of the affine plane) by the sign of the discriminant of . In other words, if we write where and put , then an isomorphism class is one of three types: ellipses (when ), parabolas (), and hyperbolas (). Of course when we admit possibly singular conic sections, we get further isomorphism classes involving some degree of degeneracy (e.g., two lines, a double line, etc.).
The distinctions between ellipse, parabola, and hyperbola are artifacts of affine geometry: if we instead consider conic sections as projective subvarieties of , then considered up to projective transformations (automorphisms of the projective plane), these distinctions evaporate and there is really only one kind of nonsingular conic section. Put differently: if we fix a representation where is a chosen “line at infinity”, then in the original classification up to affine transformations, i.e., the subgroup of projective transformations which take to itself, ellipses are those conic sections which do not intersect , parabolas are those which intersect in a double point, and hyperbolas are those which intersect in two points. By enlarging to the group of all projective transformations, we can move to a line which does intersect an “ellipse” in two points, making it a “hyperbola” with respect to the new coordinate system, etc.
Considered in terms of projective geometry, all pointed1 nonsingular conic sections are isomorphic and can be identified explicitly with a projective line by means of a stereographic projection.
Geometrically, if is the chosen basepoint of and is a line not incident to , then for any other point of the unique line incident to and intersects in exactly one point, denoted . (Here to be the intersection of the tangent to at with ; this can be considered the basepoint of .) In the opposite direction, to each point of , the line intersects in and (since a quadratic with one root will also have another root) another point (which might be the same as ; this happens precisely when is the tangent at ); this gives the inverse . In this way we obtain an isomorphism of subvarieties.
Working over an algebraically closed field , where every nonsingular conic has a point, we may conclude that is isomorphic (as a projective variety) to . Hence is a curve of genus .
Working over an algebraically closed field (let us assume the characteristic is not ), all nondegenerate quadratic forms on a vector space are isomorphic and we may fix one as standard. For example, for , we may fix attention on the quadratic form , which determines a conic section and an accompanying nondegenerate symmetric bilinear form .
Projective duality relative to is the projectivization of linear duality with respect to , which takes a linear subspace to its orthogonal dual . We note that the orthogonal dual is an involution that takes joins of subspaces to meets and vice-versa. This construction descends through the quotient to give an operation that takes points in (lines in ) to lines in (hyperplanes in ), and vice-versa, and moreover takes a join of two distinct points (the line incident to them) to the meet of their dual lines (the point of their intersection).
This duality may be visualized thus: given a nondegenerate conic and a point off of , draw the two lines incident to that are tangent to , and pass to the line incident to the tangent points. (This is easier to visualize by imagining and considering a point exterior to say an ellipse .) This defines the line that is projectively dual to (dual with respect to the conic ); if is on , then the same procedure works by considering the two tangent points as infinitesimally close to , so that the line between them is the tangent line at : the projective dual of on is its tangent line (this is the case where the line in corresponding to is isotropic with respect to the bilinear form).
The entire procedure can be reversed and gives an anti-involution on the poset of flats of , interchanging points and lines and interchanging meets and joins.
More generally, projective duality can be described in terms of an orthogonality map , where is the linear dual of , which maps a subvariety to a corresponding dual “envelope” subvariety. This is explored in GKZ. The role of the conic section (or generally a conic hypersurface) is simply to set up an explicit self-duality .
(…)
We took care to say pointed: note that depending on the field , there might not even be a solution point on the conic , even if is by all rights nonsingular. For example, consider over . Of course we can relax again if is algebraically closed. ↩
Last revised on January 3, 2015 at 21:51:44. See the history of this page for a list of all contributions to it.