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 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(\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.

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