Quadric (projective geometry)

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

(Q1) $\rho(\lambda\vec x)=\lambda^2\rho(\vec x )$ for any $\lambda\in K$ 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 symmetric.

In case of $\operatorname{char}K\ne2$ 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

$\rho(\vec x)=\sum_{1=i\le k}^{n} a_{ik}x_ix_k\text{ with }a_{ik}:= f(\vec e_i,\vec e_k)\text{ for }i\ne k\text{ and }a_{ik}:= \rho(\vec e_i)\text{ for }i=k$ and

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

For example:

$n=3,\ \rho(\vec x)=x_1x_2-x^2_3, \ f(\vec x,\vec y)=x_1y_2+x_2y_1-2x_3y_3.$

Definition and properties of a quadric

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

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

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 subspace generated by $\vec x$),

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

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: $\mathcal Q\ne \emptyset$.

Example: For $\rho(\vec x)=x_1x_2-x^2_3$ 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 $\operatorname{char}K\ne2$ 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. $\emptyset=\mathcal S\ne \mathcal R$.

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 $\operatorname{char}K\ne2$ 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: $i\le \frac{n+1}{2}$.

Let be $\mathcal Q$ a non-degenerate quadric in $\mathfrak P_n(K), n\ge 2$, 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 $\rho(\vec x)=x_1x_2-x^2_3$ 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 $\rho(\vec x)=x^2_1+a_0x_1x_2+b_0x^2_2-x_3x_4$ gives rise to a non-degenerate quadric $\mathcal Q$ in $\mathfrak P_3(K)$.

c) In $\mathfrak P_3(K)$ the quadratic form $\rho(\vec x)=x_1x_2+x_3x_4$ 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 from usual quadrics. The reason is the following statement.

Theorem: A division ring $K$ is commutative if and only if any equation $x^2+ax+b=0, \ a,b \in K$ 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