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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

## Contents

#### References

 [1] S.P. Finikov, "Analytic geometry" , Moscow (1952) (In Russian) [2] N.V. Efimov, "A short course of analytical geometry" , Moscow (1967) (In Russian)

#### References

 [a1] H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1965) pp. 65–94 [a2] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963) [a3] M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French) [a4] D. Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination" , Chelsea (1952) (Translated from German) [a5] H.F. Baker, "Principles of geometry" , 3. Solid gometry , Cambridge Univ. Press (1961)

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 [3]. Intersections of two quadrics are studied in [2], of three in [4].

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 [1].

The study of quadrics over non-closed fields is closely related to the arithmetic of quadratic forms.

#### References

 [1] D. Mumford, "Varieties defined by quadratic equations" , Questions on algebraic varieties, C.I.M.E. Varenna, 1969 , Cremonese (1970) pp. 29–100 [2] M. Reid, "The complete intersection of two or more quadrics" (1972) (Ph.D. Thesis) [3] V.S. [V.S. Ryaben'kii] Rjabenki, A.F. [A.F. Filippov] Filipov, "Über die stabilität von Differenzgleichungen" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian) [4] A.N. Tyurin, "On the intersection of quadrics" Russian Math. Surveys , 30 : 6 (1975) pp. 51–106 Uspekhi Mat. Nauk , 30 : 6 (1975) pp. 51–99

V.A. Iskovskikh