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A quadric is a [[Surface of the second order|surface of the second order]]. In a three-dimensional (projective, affine or Euclidean) space a quadric is a set of points whose homogeneous coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q0762201.png" /> (with respect to a projective, affine or Cartesian system of coordinates) satisfy a homogeneous equation of degree two:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q0762202.png" /></td> </tr></table>
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A quadric is a [[Surface of the second order|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
 
The bilinear symmetric form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q0762203.png" /></td> </tr></table>
+
$$
 +
\Phi ( x , \widetilde{x}  )  = \
 +
\sum _ {i , j = 0 } ^ { 3 }  a _ {ij} x _ {i} \widetilde{x}  _ {j}  $$
  
is called the polar form relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q0762204.png" />. Two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q0762205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q0762206.png" /> at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q0762207.png" /> are called conjugate points with respect to the quadric. If the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q0762208.png" /> intersects the quadric at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q0762209.png" /> and the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622010.png" /> are conjugate to each other with respect to the quadric, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622012.png" /> form a [[Harmonic quadruple|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622013.png" /> with respect to a quadric is called the polar of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622014.png" /> with respect to the quadric. A tangent plane to a quadric is the polar of the point of contact. The polar of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622015.png" /> is defined by the linear equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622016.png" /> with respect to the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622017.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622018.png" />, then the polar of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622019.png" /> is a plane; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622020.png" />, then the polar of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622021.png" /> is the whole space. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622022.png" /> belongs to the quadric and is called a singular point of it. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622023.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622024.png" />, then a quadric is a [[Cone|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622025.png" />, then the quadric splits into a pair of planes (real or imaginary), intersecting in a line consisting of its singular points. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622026.png" />, 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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622027.png" />. 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.
+
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|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|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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.P. Finikov, "Analytic geometry" , Moscow (1952) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N.V. Efimov, "A short course of analytical geometry" , Moscow (1967) (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.P. Finikov, "Analytic geometry" , Moscow (1952) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N.V. Efimov, "A short course of analytical geometry" , Moscow (1967) (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
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A quadric in algebraic geometry is a projective [[Algebraic variety|algebraic variety]] defined by a homogeneous quadratic equation
 
A quadric in algebraic geometry is a projective [[Algebraic variety|algebraic variety]] defined by a homogeneous quadratic equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622028.png" /></td> </tr></table>
+
$$
 +
\sum _ {i , j = 0 } ^ { n }
 +
a _ {ij} x _ {i} x _ {j}  = 0
 +
$$
  
in the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622029.png" /> over a ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622030.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622031.png" /> be a quadric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622032.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622033.png" /> be the set of its singular points. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622034.png" /> is the empty set if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622036.png" /> is the rank of the corresponding [[Quadratic form|quadratic form]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622037.png" /> is non-empty, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622038.png" /> is a [[Cone|cone]] over a non-degenerate quadric of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622039.png" /> whose vertex is the projective subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622040.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622041.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622042.png" />. All quadrics with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622043.png" /> are projectively equivalent to the quadric
+
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|quadratic form]]. If $  s ( Q) $
 +
is non-empty, then $  Q $
 +
is a [[Cone|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622044.png" /></td> </tr></table>
+
$$
 +
\sum _ { i= } 0 ^ { r- }  1 x _ {i}  ^ {2}  = 0 .
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622045.png" /> be empty and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622046.png" /> be a linear subspace of maximal dimension (it is called a generator of the quadric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622047.png" />). Then
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622048.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622049.png" />;
+
a) if $  \mathop{\rm dim}  Q = 2 m $,  
 +
then $  \mathop{\rm dim}  E = m $;
  
b) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622050.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622051.png" />.
+
b) if $  \mathop{\rm dim}  Q = 2 m + 1 $,  
 +
then $  \mathop{\rm dim}  E = m $.
  
Furthermore, the family of all subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622052.png" /> of maximal dimension on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622053.png" /> is a closed non-singular subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622054.png" /> of the [[Grassmann manifold|Grassmann manifold]] of subspaces of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622055.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622056.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622060.png" />, are non-intersecting non-singular irreducible rational varieties of the same dimension
+
Furthermore, the family of all subspaces $  E $
 +
of maximal dimension on $  Q $
 +
is a closed non-singular subset $  G $
 +
of the [[Grassmann manifold|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622061.png" /></td> </tr></table>
+
$$
 +
\left ( \begin{array}{c}
 +
m+ 1 \\
 +
2
 +
\end{array}
 +
\right ) ,
 +
$$
  
while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622063.png" /> belong to the same component if and only if
+
while $  E $
 +
and $  E  ^  \prime  $
 +
belong to the same component if and only if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622064.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm dim} ( E \cap E  ^  \prime  )  \equiv \
 +
\mathop{\rm dim}  E  (  \mathop{\rm mod}  2 ) .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622065.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622066.png" /> is a non-singular irreducible rational variety of dimension
+
If $  \mathop{\rm dim}  Q = 2 m + 1 $,  
 +
then $  G $
 +
is a non-singular irreducible rational variety of dimension
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622067.png" /></td> </tr></table>
+
$$
 +
\left ( \begin{array}{c}
 +
m+ 2 \\
 +
2
 +
\end{array}
 +
\right ) .
 +
$$
  
In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622068.png" /> is empty and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622070.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622071.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622072.png" /> (where Pic denotes the [[Picard group|Picard group]]).
+
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|Picard group]]).
  
Every quadric is rational: A birational isomorphism of a quadric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622073.png" /> with a projective space is determined by stereographic projection of the quadric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622074.png" /> from some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622076.png" />. Varieties that are complete intersections of quadrics have been studied from the point of view of birational geometry [[#References|[3]]]. Intersections of two quadrics are studied in [[#References|[2]]], of three in [[#References|[4]]].
+
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 [[#References|[3]]]. Intersections of two quadrics are studied in [[#References|[2]]], of three in [[#References|[4]]].
  
Any projective variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622077.png" /> can be imbedded in a projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622078.png" /> (for sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076220/q07622079.png" />) so that its image is the intersection (generally incomplete) of the quadrics containing it [[#References|[1]]].
+
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 [[#References|[1]]].
  
 
The study of quadrics over non-closed fields is closely related to the arithmetic of quadratic forms.
 
The study of quadrics over non-closed fields is closely related to the arithmetic of quadratic forms.
Line 62: Line 166:
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Griffiths, S. Harris, "Principles of algebraic curves" , Wiley (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , '''II''' , Cambridge Univ. Press (1952) {{MR|0048065}} {{ZBL|0048.14502}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H. Lenz, "Vorlesungen über projektive Geometrie" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1965) {{MR|0199772}} {{ZBL|0134.16203}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> G. Pickert, "Analytische Geometrie" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1953) {{MR|0058222}} {{ZBL|0051.37502}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> R. Donagi, "Group law on the intersection of two quadrics" ''Ann. Sc. Norm. Sup. Pisa Ser. IV'' , '''7''' (1980) pp. 217–240 {{MR|0581142}} {{ZBL|0457.14023}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> I.Y. Mérirdol, "Théorème de Torelli affine pour les intersections de deux quadriques" ''Invent. Math.'' , '''80''' (1985) pp. 375–416</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Griffiths, S. Harris, "Principles of algebraic curves" , Wiley (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , '''II''' , Cambridge Univ. Press (1952) {{MR|0048065}} {{ZBL|0048.14502}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H. Lenz, "Vorlesungen über projektive Geometrie" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1965) {{MR|0199772}} {{ZBL|0134.16203}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> G. Pickert, "Analytische Geometrie" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1953) {{MR|0058222}} {{ZBL|0051.37502}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> R. Donagi, "Group law on the intersection of two quadrics" ''Ann. Sc. Norm. Sup. Pisa Ser. IV'' , '''7''' (1980) pp. 217–240 {{MR|0581142}} {{ZBL|0457.14023}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> I.Y. Mérirdol, "Théorème de Torelli affine pour les intersections de deux quadriques" ''Invent. Math.'' , '''80''' (1985) pp. 375–416</TD></TR></table>

Latest revision as of 08:08, 6 June 2020


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)

Comments

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

Comments

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
How to Cite This Entry:
Quadric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quadric&oldid=23942
This article was adapted from an original article by V.S. Malakhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article