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.

(Q1) ${\displaystyle \rho (\lambda {\vec {x}})=\lambda ^{2}\rho ({\vec {x}})}$ for any ${\displaystyle \lambda \in K}$ and ${\displaystyle {\vec {x}}\in {\mathcal {V}}(K)}$.
(Q2) ${\displaystyle f({\vec {x}},{\vec {y}}):=\rho ({\vec {x}}+{\vec {y}})-\rho ({\vec {x}})-\rho ({\vec {y}})}$ is a bilinear form.

is called quadratic form. The bilinear form ${\displaystyle f}$ is symmetric.

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

For example:

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

## Definition and properties of a quadric

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

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

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

the set of lines.

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

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

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

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

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

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

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

a) ${\displaystyle g\cap {\mathcal {Q}}=\emptyset }$ and ${\displaystyle g}$ is called exterior line or
b) ${\displaystyle g\subset {\mathcal {Q}}}$ and ${\displaystyle g}$ is called tangent line or
b′) ${\displaystyle |g\cap {\mathcal {Q}}|=1}$ and ${\displaystyle g}$ is called tangent line or
c) ${\displaystyle |g\cap {\mathcal {Q}}|=2}$ and ${\displaystyle g}$ is called secant line.

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

Lemma:

a) ${\displaystyle {\mathcal {R}}:=\{P\in {\mathcal {P}}\mid P^{\perp }={\mathcal {P}}\}}$ is a flat (projective subspace). ${\displaystyle {\mathcal {R}}}$ is called f-radical of quadric ${\displaystyle {\mathcal {Q}}}$.
b) ${\displaystyle {\mathcal {S}}:={\mathcal {R}}\cap {\mathcal {Q}}}$ is a flat. ${\displaystyle {\mathcal {S}}}$ is called singular radical or ${\displaystyle \rho }$-radical of ${\displaystyle {\mathcal {Q}}}$.
c) In case of ${\displaystyle \operatorname {char} K\neq 2}$ we have ${\displaystyle {\mathcal {R}}={\mathcal {S}}}$.

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

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

A quadric is a rather homogeneous object:

The linear mapping

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

Remark:

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

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

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

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

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

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

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

Example:

a) Quadric ${\displaystyle {\mathcal {Q}}}$ in ${\displaystyle {\mathfrak {P}}_{2}(K)}$ with form ${\displaystyle \rho ({\vec {x}})=x_{1}x_{2}-x_{3}^{2}}$ is non-degenerate with index 1.
b) If polynomial ${\displaystyle q(\xi )=\xi ^{2}+a_{0}\xi +b_{0}}$ is irreducible over ${\displaystyle K}$ the quadratic form ${\displaystyle \rho ({\vec {x}})=x_{1}^{2}+a_{0}x_{1}x_{2}+b_{0}x_{2}^{2}-x_{3}x_{4}}$ gives rise to a non-degenerate quadric ${\displaystyle {\mathcal {Q}}}$ in ${\displaystyle {\mathfrak {P}}_{3}(K)}$.
c) In ${\displaystyle {\mathfrak {P}}_{3}(K)}$ the quadratic form ${\displaystyle \rho ({\vec {x}})=x_{1}x_{2}+x_{3}x_{4}}$ 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 from usual quadrics. The reason is the following statement.

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