Quadric (projective geometry)

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In projective geometry, a quadric is the set of points of a projective space where a certain quadratic form on the homogeneous coordinates becomes zero. We shall restrict ourself to the case of finite-dimensional projective spaces.

Quadratic forms

Let be $K$ a field and ${\mathcal V}(K)$ a vector space over $K$ . A mapping $\rho$ from ${\mathcal V}(K)$ to $K$ such that

(Q1) $\rho (x{\vec x})=x^{2}\rho ({\vec x})$ for any $x\in K$ and ${\vec x}\in {\mathcal V}(K)$ .

(Q2) $f({\vec x},{\vec y}):=\rho ({\vec x}+{\vec y})-\rho ({\vec x})-\rho ({\vec y})$ is a bilinear form.

is called quadratic form. (the bilinear form $f$ is even symmetric!)

In case of $\operatorname {char}K\neq 2$ we have $f({\vec x},{\vec x})=2\rho ({\vec x})$ , i.e. $f$ and $\rho$ are mutually determined in a unique way.
In case of $\operatorname {char}K=2$ we have always $f({\vec x},{\vec x})=0$ , i.e. $f$ is symplectic.

For ${\mathcal V}(K)=K^{n}$ and ${\vec x}=\sum _{{i=1}}^{{n}}x_{i}{\vec e}_{i}$ ( $\{{\vec e}_{1},\ldots ,{\vec e}_{n}\}$ is a base of ${\mathcal V}(K)$ ) $\rho$ has the form

$\rho ({\vec x})=\sum _{{1=i\leq k}}^{{n}}a_{{ik}}x_{i}x_{k}{\text{ with }}a_{{ik}}:=f({\vec e}_{i},{\vec e}_{k}){\text{ for }}i\neq k{\text{ and }}a_{{ik}}:=\rho ({\vec e}_{i}){\text{ for }}i=k$ and

$f({\vec x},{\vec y})=\sum _{{1=i\leq k}}^{{n}}a_{{ik}}(x_{i}y_{k}+x_{k}y_{i})$ .

For example:

$n=3,\ \rho ({\vec x})=x_{1}x_{2}-x_{3}^{2},\ f({\vec x},{\vec y})=x_{1}y_{2}+x_{2}y_{1}-2x_{3}y_{3}.$

Definition and properties of a quadric

Below let $K$ be a field, $2\leq n\in \mathbb{N}$ , and ${\mathfrak P}_{n}(K)=({{\mathcal P}},{{\mathcal G}},\in )$ the $n$ -dimensional projective space over $K$ , i.e.

${{\mathcal P}}=\{\langle {\vec x}\rangle \mid {\vec 0}\neq {\vec x}\in V_{{n+1}}(K)\},$

the set of points. ( $V_{{n+1}}(K)$ is a $(n + 1)$ -dimensional vector space over the field $K$ and $\langle {\vec x}\rangle$ is the 1-dimensional ${\vec x}$ ),

${{\mathcal G}}=\{\{\langle {\vec x}\rangle \in {{\mathcal P}}\mid {\vec x}\in U\}\mid U{\text{ 2-dimensional subspace of }}V_{{n+1}}(K)\},$

the set of lines.

Additionally let be $\rho$ a quadratic form on vector space $V_{{n+1}}(K)$ . A point $\langle {\vec x}\rangle \in {{\mathcal P}}$ is called singular if $\rho ({\vec x})=0$ . The set

${\mathcal Q}=\{\langle {\vec x}\rangle \in {{\mathcal P}}\mid \rho ({\vec x})=0\}$

of singular points of $\rho$ is called quadric (with respect to the quadratic form $\rho$ ). For point $P=\langle {\vec p}\rangle \in {{\mathcal P}}$ the set

$P^{\perp }:=\{\langle {\vec x}\rangle \in {{\mathcal P}}\mid f({\vec p},{\vec x})=0\}$

is called polar space of $P$ (with respect to $\rho$ ). Obviously $P^{\perp }$ is either a hyperplane or ${{\mathcal P}}$ .

For the considerations below we assume: ${\mathcal Q}\neq \emptyset$ .

Example: For $\rho ({\vec x})=x_{1}x_{2}-x_{3}^{2}$ we get a conic in ${\mathfrak P}_{2}(K)$ .

For the intersection of a line with a quadric ${\mathcal Q}$ we get:

Lemma: For a line $g$ (of $P_{n}(K)$ ) the following cases occur:

a) $g\cap {\mathcal Q}=\emptyset$ and $g$ is called exterior line or

b) $g\subset {\mathcal Q}$ and $g$ is called tangent line or

b′) $|g\cap {\mathcal Q}|=1$ and $g$ is called tangent line or

c) $|g\cap {\mathcal Q}|=2$ and $g$ is called secant line.

Lemma: A line $g$ through point $P\in {\mathcal Q}$ is a tangent line if and only if $g\subset P^{\perp }$ .

Lemma:

a) ${\mathcal R}:=\{P\in {{\mathcal P}}\mid P^{\perp }={\mathcal P}\}$ is a flat (projective subspace). ${\mathcal R}$ is called f-radical of quadric ${\mathcal Q}$ .

b) ${\mathcal S}:={\mathcal R}\cap {\mathcal Q}$ is a flat. ${\mathcal S}$ is called singular radical or $\rho$ -radical of ${\mathcal Q}$ .

c) In case of $\operatorname {char}K\neq 2$ we have ${\mathcal R}={\mathcal S}$ .

A quadric is called non-degenerate if ${\mathcal S}=\emptyset$ .

Remark: An oval conic is a non-degenerate quadric. In case of $\operatorname {char}K=2$ its knot is the f-radical, i.e. $\emptyset ={\mathcal S}\neq {\mathcal R}$ .

A quadric is a rather homogeneous object:

Lemma: For any point $P\in {{\mathcal P}}\setminus ({\mathcal Q}\cup {{\mathcal R}})$ there exists an involutorial central collineation $\sigma _{P}$ with center $P$ and $\sigma _{P}({\mathcal Q})={\mathcal Q}$ .

Proof: Due to $P\in {{\mathcal P}}\setminus ({\mathcal Q}\cup {{\mathcal R}})$ the polar space $P^{\perp }$ is a hyperplane.

The linear mapping

$\varphi :{\vec x}\rightarrow {\vec x}-{\frac {f({\vec p},{\vec x})}{\rho ({\vec p})}}{\vec p}$

induces an involutorial central collineation with axis $P^{\perp }$ and centre $P$ which leaves ${\mathcal Q}$ invariant.
In case of $\operatorname {char}K\neq 2$ mapping $\varphi$ gets the familiar shape $\varphi :{\vec x}\rightarrow {\vec x}-2{\frac {f({\vec p},{\vec x})}{f({\vec p},{\vec p})}}{\vec p}$ with $\varphi ({\vec p})=-{\vec p}$ and $\varphi ({\vec x})={\vec x}$ for any $\langle {\vec x}\rangle \in P^{\perp }$ .

Remark:

a) The image of an exterior, tangent and secant line, respectively, by the involution $\sigma _{P}$ of the Lemma above is an exterior, tangent and secant line, respectively.

b) ${{\mathcal R}}$ is pointwise fixed by $\sigma _{P}$ .

Let be $\Pi ({\mathcal Q})$ the group of projective collineations of ${\mathfrak P}_{n}(K)$ which leaves ${\mathcal Q}$ invariant. We get

Lemma: $\Pi ({\mathcal Q})$ operates transitively on ${\mathcal Q}\setminus {{\mathcal R}}$ .

A subspace ${\mathcal U}$ of ${\mathfrak P}_{n}(K)$ is called $\rho$ -subspace if ${\mathcal U}\subset {\mathcal Q}$ (for example: points on a sphere or lines on a hyperboloid (s. below)).

Lemma: Any two maximal $\rho$ -subspaces have the same dimension $m$ .

Let be $m$ the dimension of the maximal $\rho$ -subspaces of ${\mathcal Q}$ . The integer $i:=m+1$ is called index of ${\mathcal Q}$ .

Theorem: (BUEKENHOUT) For the index $i$ of a non-degenerate quadric ${\mathcal Q}$ in ${\mathfrak P}_{n}(K)$ the following is true: $i\leq {\frac {n+1}{2}}$ .

Let be ${\mathcal Q}$ a non-degenerate quadric in ${\mathfrak P}_{n}(K),n\geq 2$ , and $i$ its index.

In case of $i=1$ quadric ${\mathcal Q}$ is called sphere (or oval conic if $n=2$ ).

In case of $i=2$ quadric ${\mathcal Q}$ is called hyperboloid (of one sheet).

Example:

a) Quadric ${\mathcal Q}$ in ${\mathfrak P}_{2}(K)$ with form $\rho ({\vec x})=x_{1}x_{2}-x_{3}^{2}$ is non-degenerate with index 1.

b) If polynomial $q(\xi )=\xi ^{2}+a_{0}\xi +b_{0}$ is irreducible over $K$ the quadratic form $\rho ({\vec x})=x_{1}^{2}+a_{0}x_{1}x_{2}+b_{0}x_{2}^{2}-x_{3}x_{4}$ gives rise to a non-degenerate quadric ${\mathcal Q}$ in ${\mathfrak P}_{3}(K)$ .

c) In ${\mathfrak P}_{3}(K)$ the quadratic form $\rho ({\vec x})=x_{1}x_{2}+x_{3}x_{4}$ gives rise to a hyperboloid.

Remark: It is not reasonable to define formally quadrics for "vector spaces" (strictly speaking, modules) over genuine skew fields (division rings). Because one would get secants bearing more than 2 points of the quadric which is totally different to usual quadrics. The reason is the following statement.

Theorem: A division ring $K$ is commutative if and only if any $x^{2}+ax+b=0,\ a,b\in K$ has at most two solutions.

There are generalizations of quadrics: quadratic sets. A quadratic set is a set of points of a projective plane/space, which bears the same geometric properties as a quadric: any line intersects a quadratic set in no or 1 or two lines or is containt in the set.

External links

Lecture Note Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes, p. 117

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