A quadric is a surface of the second order. In a three-dimensional (projective, affine or Euclidean) space a quadric is a set of points whose homogeneous coordinates (with respect to a projective, affine or Cartesian system of coordinates) satisfy a homogeneous equation of degree two:
The bilinear symmetric form
is called the polar form relative to . Two points , at which are called conjugate points with respect to the quadric. If the line intersects the quadric at the points and the points are conjugate to each other with respect to the quadric, then and form a harmonic quadruple. The points of a quadric and only these are self-conjugate. A line all points of which lie on a quadric is called a generator of the quadric. By the pole of a given plane with respect to a quadric is meant the point that is conjugate to every point of this plane. The set of points in the space that are conjugate to a given point with respect to a quadric is called the polar of with respect to the quadric. A tangent plane to a quadric is the polar of the point of contact. The polar of a point is defined by the linear equation with respect to the coordinates . If , then the polar of is a plane; if , then the polar of is the whole space. In this case belongs to the quadric and is called a singular point of it. If , then the quadric has no singular points and is called a non-degenerate quadric. In projective space this is an imaginary ovaloid, a real ovaloid or a ruled quadric. A non-degenerate quadric determines a correlation (or polarity), i.e. a bijective mapping from the set of points of projective space onto the set of planes. A ruled non-degenerate quadric has two distinct families of generators, distributed on the quadric so that any two lines of the same family are non-intersecting, while two lines of different families intersect at one point. If , then a quadric is a cone (real or imaginary) with vertex at the unique singular point. A real cone has a single family of generators, passing through its vertex. If , then the quadric splits into a pair of planes (real or imaginary), intersecting in a line consisting of its singular points. If , then a quadric is a double real plane consisting of singular points. The affine properties of a quadric are distinguished by its behaviour with respect to the plane at infinity, . For example, an ellipsoid (hyperboloid, paraboloid) is a non-degenerate quadric that does not intersect (does intersect, is tangent to) the plane at infinity. The centre of a quadric is the pole of the plane at infinity, a diameter is a line through the centre.
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A quadric in algebraic geometry is a projective algebraic variety defined by a homogeneous quadratic equation
in the projective space over a ground field .
Suppose further that the ground field is algebraically closed and has characteristic not equal to 2. Let be a quadric in and let be the set of its singular points. Then is the empty set if and only if , where is the rank of the corresponding quadratic form. If is non-empty, then is a cone over a non-degenerate quadric of dimension whose vertex is the projective subspace in of dimension . All quadrics with are projectively equivalent to the quadric
Let be empty and let be a linear subspace of maximal dimension (it is called a generator of the quadric ). Then
a) if , then ;
b) if , then .
Furthermore, the family of all subspaces of maximal dimension on is a closed non-singular subset of the Grassmann manifold of subspaces of dimension in . If , , , , are non-intersecting non-singular irreducible rational varieties of the same dimension
while and belong to the same component if and only if
If , then is a non-singular irreducible rational variety of dimension
In case is empty and , ; if , then (where Pic denotes the Picard group).
Every quadric is rational: A birational isomorphism of a quadric with a projective space is determined by stereographic projection of the quadric from some point , . Varieties that are complete intersections of quadrics have been studied from the point of view of birational geometry . Intersections of two quadrics are studied in , of three in .
Any projective variety can be imbedded in a projective space (for sufficiently large ) so that its image is the intersection (generally incomplete) of the quadrics containing it .
The study of quadrics over non-closed fields is closely related to the arithmetic of quadratic forms.
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Quadric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quadric&oldid=11769