# 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 a field and a vector space over . A mapping from to such that

**(Q2)**is a bilinear form.

is called **quadratic form**. The bilinear form is symmetric*.*

In case of we have , i.e. and are mutually determined in a unique way.

In case of we have always , i.e. is
*symplectic*.

For and ( is a base of ) has the form

For example:

## Definition and properties of a quadric

Below let be a field, , and the Template:Mvar-dimensional projective space over , i.e.

the *set of points*. ( is a (*n* + 1)-dimensional vector space over the field and is the 1-dimensional subspace generated by ),

the *set of lines*.

Additionally let be a quadratic form on vector space . A point is called *singular* if . The set

of singular points of is called *quadric* (with respect to the quadratic form ). For point the set

is called *polar space* of (with respect to ).
Obviously is either a hyperplane or .

For the considerations below we assume: .

**Example:**
For we get a conic in .

For the intersection of a line with a quadric we get:

**Lemma:**
For a line (of ) the following cases occur:

- a) and is called
*exterior line*or - b) and is called
*tangent line*or - b′) and is called
*tangent line*or - c) and is called
*secant line*.

**Lemma:**
A line through point is a tangent line if and only if
.

**Lemma:**

- a) is a flat (projective subspace). is called
*f-radical*of quadric . - b) is a flat. is called
*singular radical*or*-radical*of . - c) In case of we have .

A quadric is called *non-degenerate* if .

**Remark:**
An oval conic is a non-degenerate quadric. In case of its knot is the f-radical, i.e. .

A quadric is a rather homogeneous object:

**Lemma:**
For any point there exists an involutorial central collineation with center and
.

**Proof:**
Due to the polar space is a hyperplane.

The linear mapping

induces an *involutorial central collineation* with axis and centre which leaves invariant.

In case of mapping gets the familiar shape with and for any .

**Remark:**

- a) The image of an exterior, tangent and secant line, respectively, by the involution of the Lemma above is an exterior, tangent and secant line, respectively.
- b) is pointwise fixed by .

Let be the group of projective collineations of which leaves invariant. We get

**Lemma:**
operates transitively on .

A subspace of is called -subspace if (for example: points on a sphere or lines on a hyperboloid (s. below)).

**Lemma:**
Any two *maximal* -subspaces have the same dimension .

Let be the dimension of the maximal -subspaces of .
The integer is called *index of* .

**Theorem: (BUEKENHOUT)**
For the index of a non-degenerate quadric in the following is
true: .

Let be a non-degenerate quadric in , and its index.

- In case of quadric is called
*sphere*(or oval conic if ). - In case of quadric is called
*hyperboloid*(of one sheet).

**Example:**

- a) Quadric in with form is non-degenerate with index 1.
- b) If polynomial is irreducible over the quadratic form gives rise to a non-degenerate quadric in .
- c) In the quadratic form 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 is commutative if and only if any equation 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.