Editing Quadric (section)

== Projective quadrics over fields ==

The definition of a projective quadric in a real projective space (see above) can be formally adapted by defining a projective quadric in an ''n''-dimensional projective space over a [[Field (mathematics)|field]]. In order to omit dealing with coordinates, a projective quadric is usually defined by starting with a quadratic form on a vector space.<ref>Beutelspacher/Rosenbaum p.158</ref>

===Quadratic form===
Let <math>K</math> be a [[Field (algebra)|field]] and <math>V</math> a [[vector space]] over <math>K</math>. A mapping <math>q</math> from <math>V</math> to <math>K</math> such that

: '''(Q1)''' <math>\;q(\lambda\vec x)=\lambda^2q(\vec x )\;</math>  for any <math>\lambda\in K</math>  and  <math>\vec x \in V</math>.

: '''(Q2)''' <math>\;f(\vec x,\vec y ):=q(\vec x+\vec y)-q(\vec x)-q(\vec y)\;</math> is a [[bilinear form]].
is called '''[[quadratic form]]'''. The bilinear form <math>f</math> is symmetric''.''

In case of <math>\operatorname{char}K\ne2</math> the bilinear form is <math>f(\vec x,\vec x)=2q(\vec x)</math>, i.e. <math>f</math> and <math>q</math> are mutually determined in a unique way.<br />
In case of <math>\operatorname{char}K=2</math> (that means: <math>1+1=0</math>) the bilinear form has the property <math>f(\vec x,\vec x)=0</math>, i.e. <math>f</math> is
''[[Symplectic vector space|symplectic]]''.

For <math>V=K^n\ </math> and <math>\ \vec x=\sum_{i=1}^{n}x_i\vec e_i\quad </math>
(<math>\{\vec e_1,\ldots,\vec e_n\} </math> is a base of <math>V</math>) <math>\ q</math> has the familiar form
: <math>
q(\vec x)=\sum_{1=i\le k}^{n} a_{ik}x_ix_k\ \text{ with }\ a_{ik}:= f(\vec e_i,\vec e_k)\ \text{ for }\ i\ne k\ \text{ and }\ a_{ii}:= q(\vec e_i)\ </math> and

: <math> f(\vec x,\vec y)=\sum_{1=i\le k}^{n} a_{ik}(x_iy_k+x_ky_i)</math>.

For example:

: <math>n=3,\quad q(\vec x)=x_1x_2-x^2_3, \quad f(\vec x,\vec y)=x_1y_2+x_2y_1-2x_3y_3\; . </math>

=== ''n''-dimensional projective space over a field ===
Let <math>K</math> be a field, <math>2\le n\in\N</math>,
:<math>V_{n+1}</math>  an {{math|(''n'' + 1)}}-[[dimension (vector space)|dimensional]] [[vector space]] over the field <math>K,</math>
:<math>\langle\vec x\rangle</math> the 1-dimensional [[linear span|subspace generated by <math>\vec 0\ne \vec x\in V_{n+1}</math>]],
: <math>{\mathcal P}=\{\langle \vec x\rangle \mid \vec x \in V_{n+1}\},\ </math> the ''set of points'' ,
: <math>{\mathcal G}=\{ \text{2-dimensional subspaces of } V_{n+1}\},\ </math> the ''set of lines''.
:<math>P_n(K)=({\mathcal P},{\mathcal G})\ </math> is the {{mvar|n}}-dimensional '''[[projective space]]''' over <math>K</math>.
:The set of points contained in a <math>(k+1)</math>-dimensional subspace of <math> V_{n+1}</math> is a ''<math>k</math>-dimensional subspace'' of  <math>P_n(K)</math>. A 2-dimensional subspace is a ''plane''.
:In case of <math>\;n>3\;</math> a <math>(n-1)</math>-dimensional subspace is called ''hyperplane''.

=== Projective quadric ===
A quadratic form <math>q</math>  on a vector space <math>V_{n+1}</math> defines a ''quadric'' <math>\mathcal Q</math> in the associated projective space <math>\mathcal P,</math> as the set of the points <math>\langle\vec x\rangle \in {\mathcal P}</math> such that <math>q(\vec x)=0</math>. That is,
: <math>\mathcal Q=\{\langle\vec x\rangle \in {\mathcal P} \mid q(\vec x)=0\}.</math>

'''Examples in <math> P_2(K)</math>.:'''<br />
'''(E1):''' For <math>\;q(\vec x)=x_1x_2-x^2_3\;</math> one obtains a [[Conic section|conic]].<br />
'''(E2):''' For <math>\;q(\vec x)=x_1x_2\;</math> one obtains the pair of lines with the equations <math>x_1=0</math> and <math>x_2=0</math>, respectively. They intersect at point <math>\langle(0,0,1)^\text{T}\rangle</math>;

For the considerations below it is assumed that <math>\mathcal Q\ne \emptyset</math>.

=== Polar space ===
For point <math>P=\langle\vec p\rangle \in {\mathcal P}</math> the set
: <math>P^\perp:=\{\langle\vec x\rangle\in {\mathcal P} \mid f(\vec p,\vec x)=0\}</math>
is called [[Duality (mathematics)#Polarities of general projective spaces|'''polar space''']] of <math>P</math> (with respect to <math>q</math>).

If <math>\;f(\vec p,\vec x)=0\;</math> for all <math>\vec x </math>, one obtains <math>P^\perp=\mathcal P</math>.

If <math>\;f(\vec p,\vec x)\ne 0\;</math> for at least one <math>\vec x </math>, the equation <math>\;f(\vec p,\vec x)=0\;</math>is a non trivial linear equation which defines a hyperplane. Hence

:<math>P^\perp</math> is either a [[hyperplane]] or <math>{\mathcal P}</math>.

=== Intersection with a line ===
For the intersection of an arbitrary line <math>g</math> with a quadric <math> \mathcal Q</math>, the following cases may occur:
:a) <math>g\cap \mathcal Q=\emptyset\;</math> and <math>g</math> is called ''exterior line''
:b) <math> g \subset \mathcal Q\; </math> and <math>g</math> is called a ''line in the quadric''
:c) <math>|g\cap \mathcal Q|=1\; </math> and <math>g</math> is called ''tangent line''
:d) <math>|g\cap \mathcal Q|=2\; </math> and <math>g</math> is called ''secant line''.

'''Proof:'''
Let <math>g</math> be a line, which intersects <math>\mathcal Q </math> at point <math>\;U=\langle\vec u\rangle\;</math> and <math> \;V= \langle\vec v\rangle\;</math> is a second point on <math>g</math>.
From <math>\;q(\vec u)=0\;</math> one obtains<br />
<math>q(x\vec u+\vec v)=q(x\vec u)+q(\vec v)+f(x\vec u,\vec v)=q(\vec v)+xf(\vec u,\vec v)\; .</math><br />
I) In case of <math>g\subset U^\perp</math> the equation <math>f(\vec u,\vec v)=0</math> holds and it is
<math>\; q(x\vec u+\vec v)=q(\vec v)\; </math> for any <math>x\in K</math>. Hence either <math>\;q(x\vec u+\vec v)=0\;</math>
for ''any'' <math>x\in K</math> or <math>\;q(x\vec u+\vec v)\ne 0\;</math> for ''any'' <math>x\in K</math>, which proves b) and b').<br />
II) In case of  <math>g\not\subset U^\perp</math> one obtains <math>f(\vec u,\vec v)\ne 0</math> and the equation
<math>\;q(x\vec u+\vec v)=q(\vec v)+xf(\vec u,\vec v)= 0\;</math> has exactly one solution <math>x</math>.
Hence: <math>|g\cap \mathcal Q|=2</math>, which proves c).

Additionally the proof shows:
:A line <math>g</math> through a point <math>P\in \mathcal Q</math> is a ''tangent'' line if and only if <math>g\subset P^\perp</math>.

=== ''f''-radical, ''q''-radical ===
In the classical cases <math>K=\R</math> or <math> \C</math> there exists only one radical, because of <math>\operatorname{char}K\ne2</math> and <math>f</math> and <math>q</math> are closely connected. In case of <math>\operatorname{char}K=2</math> the quadric <math>\mathcal Q</math> is not determined by <math>f</math> (see above) and so one has to deal with two radicals:

:a) <math>\mathcal R:=\{P\in{\mathcal P} \mid P^\perp=\mathcal P\}</math> is a projective subspace. <math>\mathcal R</math> is called '''''f''-radical''' of quadric <math>\mathcal Q</math>.
:b) <math>\mathcal S:=\mathcal R\cap\mathcal Q</math> is called '''singular radical''' or ''<math>q</math>-radical'' of <math>\mathcal Q</math>.
:c) In case of <math>\operatorname{char}K\ne2</math> one has <math>\mathcal R=\mathcal S</math>.

A quadric is called '''non-degenerate''' if <math>\mathcal S=\emptyset</math>.

'''Examples in <math> P_2(K)</math>''' (see above):<br />
'''(E1):''' For <math>\;q(\vec x)=x_1x_2-x^2_3\;</math> (conic) the bilinear form is
<math>f(\vec x,\vec y)=x_1y_2+x_2y_1-2x_3y_3\; . </math><br />
In case of <math>\operatorname{char}K\ne2</math> the polar spaces are never <math>\mathcal P</math>. Hence <math>\mathcal R=\mathcal S=\empty</math>.<br />
In case of <math>\operatorname{char}K=2</math> the bilinear form is reduced to
<math>f(\vec x,\vec y)=x_1y_2+x_2y_1\; </math> and <math>\mathcal R=\langle(0,0,1)^\text{T}\rangle\notin \mathcal Q</math>. Hence <math>\mathcal R\ne \mathcal S=\empty \; .</math>
In this case the ''f''-radical is the common point of all tangents, the so called ''knot''.<br />
In both cases <math> S=\empty</math> and the quadric (conic) ist ''non-degenerate''.<br />
'''(E2):''' For <math>\;q(\vec x)=x_1x_2\;</math> (pair of lines) the bilinear form is <math>f(\vec x,\vec y)=x_1y_2+x_2y_1\;  </math> and <math>\mathcal R=\langle(0,0,1)^\text{T}\rangle=\mathcal S\; ,</math> the intersection point. <br />
In this example the quadric is ''degenerate''.

=== Symmetries ===
A quadric is a rather homogeneous object:

:For any point <math>P\notin \mathcal Q\cup {\mathcal R}\;</math> there exists an [[Involution (mathematics)|involutorial]] central [[collineation]] <math>\sigma_P</math> with center <math>P</math> and <math>\sigma_P(\mathcal Q)=\mathcal Q</math>.

'''Proof:'''
Due to <math>P\notin \mathcal Q\cup {\mathcal R}</math> the polar space <math>P^\perp</math> is a hyperplane.

The linear mapping

: <math>\varphi: \vec x \rightarrow \vec x-\frac{f(\vec p,\vec x)}{q(\vec p)}\vec p</math>

induces an ''involutorial central collineation'' <math>\sigma_P</math> with axis <math>P^\perp</math> and centre <math>P</math> which leaves <math>\mathcal Q</math> invariant.<br />
In the case of <math>\operatorname{char}K\ne2</math>, the mapping <math>\varphi</math> produces the [[Reflection (mathematics)|familiar shape]] <math>\; \varphi: \vec x \rightarrow \vec x-2\frac{f(\vec p,\vec x)}{f(\vec p,\vec p)}\vec p\; </math> with <math>\; \varphi(\vec p)=-\vec p</math> and <math>\; \varphi(\vec x)=\vec x\; </math> for any <math>\langle\vec x\rangle \in P^\perp</math>.

'''Remark:'''
:a) An exterior line, a tangent line or a secant line is mapped by the involution <math>\sigma_P</math> on an exterior, tangent and secant line, respectively.
:b) <math>{\mathcal R}</math> is pointwise fixed by <math>\sigma_P</math>.

===''q''-subspaces and index of a quadric ===
A subspace <math>\;\mathcal U\;</math> of <math>P_n(K)</math> is called <math>q</math>-subspace if <math>\;\mathcal U\subset\mathcal Q\;</math>

For example: points on a sphere or [[ruled surface|lines on a hyperboloid]] (s. below).

:Any two ''maximal'' <math>q</math>-subspaces have the same dimension <math>m</math>.<ref>Beutelpacher/Rosenbaum, p.139</ref>

Let be <math>m</math> the dimension of the maximal <math>q</math>-subspaces of <math>\mathcal Q</math> then
:The integer <math>\;i:=m+1\;</math> is called '''index''' of <math>\mathcal Q</math>.

'''Theorem: (BUEKENHOUT)<ref>F. Buekenhout: ''Ensembles Quadratiques des Espace Projective'', Math. Teitschr. 110 (1969), p. 306-318.</ref>'''
:For the index <math>i</math> of a non-degenerate quadric <math>\mathcal Q</math> in <math>P_n(K)</math> the following is true:
::<math>i\le \frac{n+1}{2}</math>.

Let be <math>\mathcal Q</math> a non-degenerate quadric in <math> P_n(K), n\ge 2</math>, and <math>i</math> its index.<br />
: In case of <math>i=1</math> quadric <math>\mathcal Q</math> is called ''sphere'' (or [[oval (projective plane)|oval]] conic if <math>n=2</math>).
: In case of <math>i=2</math> quadric <math>\mathcal Q</math> is called ''hyperboloid'' (of one sheet).

'''Examples:'''
:a) Quadric <math>\mathcal Q</math> in <math>P_2(K)</math> with form <math>\;q(\vec x)=x_1x_2-x^2_3\;</math> is non-degenerate with index 1.
:b) If polynomial <math>\;p(\xi)=\xi^2+a_0\xi+b_0\;</math> is [[Irreducible polynomial|irreducible]] over <math>K</math> the quadratic form <math>\;q(\vec x)=x^2_1+a_0x_1x_2+b_0x^2_2-x_3x_4\;</math> gives rise to a non-degenerate quadric <math>\mathcal Q</math> in <math>P_3(K)</math> of index 1 (sphere). For example: <math>\;p(\xi)=\xi^2+1\;</math> is irreducible over <math>\R</math> (but not over <math>\C</math> !).
:c) In <math>P_3(K)</math> the quadratic form <math>\;q(\vec x)=x_1x_2+x_3x_4\;</math> generates a ''hyperboloid''.

=== Generalization of quadrics: quadratic sets ===
It is not reasonable to formally extend the definition of quadrics to spaces over genuine skew fields (division rings). Because one would obtain secants bearing more than 2 points of the quadric which is totally different from ''usual'' quadrics.<ref>R. [[Rafael Artzy|Artzy]]: ''The Conic <math>y=x^2</math> in Moufang Planes'', Aequat.Mathem. 6 (1971), p. 31-35</ref><ref>E. Berz: ''Kegelschnitte in Desarguesschen Ebenen'', Math. Zeitschr. 78 (1962), p. 55-8</ref><ref>external link E. Hartmann: ''Planar Circle Geometries'', p. 123</ref> The reason is the following statement.

:A [[division ring]] <math>K</math> is [[commutative ring|commutative]] if and only if any [[quadratic equation|equation]] <math>x^2+ax+b=0, \ a,b \in K</math>, has at most two solutions.

There are ''generalizations'' of quadrics: [[quadratic set]]s.<ref>Beutelspacher/Rosenbaum: p. 135</ref> A quadratic set is a set of points of a projective space with the same geometric properties as a quadric: every line intersects a quadratic set in at most two points or is contained in the set.