Quadric

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 $ x _ {0} , x _ {1} , x _ {2} , x _ {3} $( with respect to a projective, affine or Cartesian system of coordinates) satisfy a homogeneous equation of degree two:

$$ F ( x) \equiv \ \sum _ {i , j = 0 } ^ { 3 } a _ {ij} x _ {i} x _ {j} = 0 ,\ \ a _ {ij} = a _ {ji} . $$

The bilinear symmetric form

$$ \Phi ( x , \widetilde{x} ) = \ \sum _ {i , j = 0 } ^ { 3 } a _ {ij} x _ {i} \widetilde{x} _ {j} $$

is called the polar form relative to $ F ( x) $. Two points $ M ^ \prime ( x _ {0} ^ \prime , x _ {1} ^ \prime , x _ {2} ^ \prime , x _ {3} ^ \prime ) $, $ M ^ {\prime\prime} ( x _ {0} ^ {\prime\prime} , x _ {1} ^ {\prime\prime} , x _ {2} ^ {\prime\prime} , x _ {3} ^ {\prime\prime} ) $ at which $ \Phi ( x ^ \prime , x ^ {\prime\prime} ) = 0 $ are called conjugate points with respect to the quadric. If the line $ M ^ \prime M ^ {\prime\prime} $ intersects the quadric at the points $ N _ {1} , N _ {2} $ and the points $ M ^ \prime , M ^ {\prime\prime} $ are conjugate to each other with respect to the quadric, then $ N _ {1} , N _ {2} $ and $ M ^ \prime , M ^ {\prime\prime} $ 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 $ M ^ \prime $ with respect to a quadric is called the polar of $ M ^ \prime $ with respect to the quadric. A tangent plane to a quadric is the polar of the point of contact. The polar of a point $ M ^ \prime $ is defined by the linear equation $ \Phi ( x , x ^ \prime ) \equiv 0 $ with respect to the coordinates $ x _ {0} , x _ {1} , x _ {2} , x _ {3} $. If $ \Phi ( x , x ^ \prime ) \not\equiv 0 $, then the polar of $ M ^ \prime $ is a plane; if $ \Phi ( x , x ^ \prime ) \equiv 0 $, then the polar of $ M ^ \prime $ is the whole space. In this case $ M ^ \prime $ belongs to the quadric and is called a singular point of it. If $ R = \mathop{\rm rank} ( a _ {ij} ) = 4 $, 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 $ R = 3 $, 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 $ R = 2 $, then the quadric splits into a pair of planes (real or imaginary), intersecting in a line consisting of its singular points. If $ R = 1 $, 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, $ x _ {0} = 0 $. 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.

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 MR1628013 MR1531918 MR0087965 MR0006835 Zbl 0909.51003 Zbl 0077.13903 Zbl 0060.32807 Zbl 68.0322.02
[a2]	H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963) MR0990644 MR0346644 MR0212646 MR1531486 MR0123930 Zbl 0181.48101 Zbl 0145.16803 Zbl 0095.34502
[a3]	M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French) MR0903026 MR0895392 MR0882916 MR0882541 Zbl 0619.53001 Zbl 0606.51001 Zbl 0606.00020
[a4]	D. Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination" , Chelsea (1952) (Translated from German) MR0046650 Zbl 0047.38806
[a5]	H.F. Baker, "Principles of geometry" , 3. Solid gometry , Cambridge Univ. Press (1961) MR0178392 MR0178393 Zbl 1206.14007 Zbl 1206.14006 Zbl 1206.14005 Zbl 1206.14004 Zbl 1206.14003 Zbl 1206.14002 Zbl 0008.21907 Zbl 0008.21906 Zbl 59.1290.03 Zbl 59.0620.04 Zbl 51.0531.07 Zbl 48.0686.01 Zbl 48.0646.08 Zbl 49.0395.03

A quadric in algebraic geometry is a projective algebraic variety defined by a homogeneous quadratic equation

$$ \sum _ {i , j = 0 } ^ { n } a _ {ij} x _ {i} x _ {j} = 0 $$

in the projective space $ P ^ {n} $ over a ground field $ k $.

Suppose further that the ground field is algebraically closed and has characteristic not equal to 2. Let $ Q $ be a quadric in $ P ^ {n} $ and let $ s ( Q) $ be the set of its singular points. Then $ s ( Q) $ is the empty set if and only if $ \mathop{\rm rk} ( Q) = n + 1 $, where $ \mathop{\rm rk} ( Q) $ is the rank of the corresponding quadratic form. If $ s ( Q) $ is non-empty, then $ Q $ is a cone over a non-degenerate quadric of dimension $ \mathop{\rm rk} ( Q) - 1 $ whose vertex is the projective subspace $ s ( Q) $ in $ P ^ {n} $ of dimension $ n - \mathop{\rm rk} ( Q) $. All quadrics with $ \mathop{\rm rk} ( Q) = r $ are projectively equivalent to the quadric

$$ \sum _ { i= } 0 ^ { r- } 1 x _ {i} ^ {2} = 0 . $$

Let $ s ( Q) $ be empty and let $ E \subset Q $ be a linear subspace of maximal dimension (it is called a generator of the quadric $ Q $). Then

a) if $ \mathop{\rm dim} Q = 2 m $, then $ \mathop{\rm dim} E = m $;

b) if $ \mathop{\rm dim} Q = 2 m + 1 $, then $ \mathop{\rm dim} E = m $.

Furthermore, the family of all subspaces $ E $ of maximal dimension on $ Q $ is a closed non-singular subset $ G $ of the Grassmann manifold of subspaces of dimension $ \mathop{\rm dim} E $ in $ P ^ {n} $. If $ \mathop{\rm dim} Q = 2 m $, $ G = G _ {1} \cup G _ {2} $, $ G _ {i} $, $ i = 1 , 2 $, are non-intersecting non-singular irreducible rational varieties of the same dimension

$$ \left ( \begin{array}{c} m+ 1 \\ 2 \end{array} \right ) , $$

while $ E $ and $ E ^ \prime $ belong to the same component if and only if

$$ \mathop{\rm dim} ( E \cap E ^ \prime ) \equiv \ \mathop{\rm dim} E ( \mathop{\rm mod} 2 ) . $$

If $ \mathop{\rm dim} Q = 2 m + 1 $, then $ G $ is a non-singular irreducible rational variety of dimension

$$ \left ( \begin{array}{c} m+ 2 \\ 2 \end{array} \right ) . $$

In case $ s ( Q) $ is empty and $ \mathop{\rm dim} Q = 2 $, $ Q \cong P ^ {1} \times P ^ {1} $; if $ \mathop{\rm dim} Q \neq 2 $, then $ \mathop{\rm Pic} ( Q) \cong \mathbf Z $( where Pic denotes the Picard group).

Every quadric is rational: A birational isomorphism of a quadric $ Q $ with a projective space is determined by stereographic projection of the quadric $ Q $ from some point $ q \in Q $, $ q \notin s ( Q) $. 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 $ X $ can be imbedded in a projective space $ P ^ {N} $( for sufficiently large $ N $) 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 MR0282975 Zbl 0198.25801
[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

References

[a1]	P. Griffiths, S. Harris, "Principles of algebraic curves" , Wiley (1978)
[a2]	W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , II , Cambridge Univ. Press (1952) MR0048065 Zbl 0048.14502
[a3]	H. Lenz, "Vorlesungen über projektive Geometrie" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1965) MR0199772 Zbl 0134.16203
[a4]	G. Pickert, "Analytische Geometrie" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1953) MR0058222 Zbl 0051.37502
[a5]	R. Donagi, "Group law on the intersection of two quadrics" Ann. Sc. Norm. Sup. Pisa Ser. IV , 7 (1980) pp. 217–240 MR0581142 Zbl 0457.14023
[a6]	I.Y. Mérirdol, "Théorème de Torelli affine pour les intersections de deux quadriques" Invent. Math. , 80 (1985) pp. 375–416