Flocks Glossary


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- A -

Arc
In a projective plane, a k-arc is a set of k points no three of which are collinear.
In general, in PG(n,q), a k-arc is a set of k points any n+1 of which form a basis for PG(n,q).
Sets of k points in PG(n,q) with the property that no three points are collinear are called k-caps. k-arcs in PG(2,q) are thus k-caps, but this terminology is never used.

- B -

Baer subplanes
In PG(2,q2) a subplane of order q is called a Baer suplane.
Baselines
The lines of intersection of the planes of a flock when projected from V = (0,0,0,1) into the carrier plane (W = 0) are called baselines. We distinguish two types of baselines:
BLT-Set
Blocking Set
In the most general terms, in a geometry if there are two types of objects, A and B, a set of objects of type A with the property that every object of type B intersects or is incident with some object in the set is called an (A) B blocking set. The classical example in a projective plane is a (punctual) line blocking set, i.e., a set of points in the plane such that every line of the plane is incident with at least one point of the set. Another example is that of a (lineal) conic blocking set, a set of lines in the plane such that every conic in the plane intersects at least one line of the set. When no modifiers are present, the classical example is assumed.
A proper (A) B blocking set is one that does not contain any element of type B. Thus, a proper blocking set is a set of points which contains no line.

- C -

C-Equivalent
Cap
A k-cap in a projective space is a set of k points, no three of which are collinear. See arc.
Carrier
Circle
Cone
Conic
Conic Blocking Set
A (lineal) Conic Blocking set is a set of lines in a plane which intersect every conic of the plane. See blocking set.
Complete cap
Coordinate functions
Critical Cone

- D -

Dual Setting

- E -

- F -

Fisher Type Flock
Flock
In PG(3,q) a flock is a set of q planes which do not pass through the point V = (0,0,0,1). Note that this is not a standard definition. Given a cone, C = C(V,S), a flock of C is a flock with the property that no two planes of the flock meet at any point of C. Since we are allowing the empty cone (where S is the empty set), any flock is the flock of some cone (perhaps only the empty cone), and this justifies our free-wheeling definition of flock.
There are several special types of flocks, only a few of which are listed below:
FTW Type Flock

- G -

Ganley type flock
Generalized Quadrangle
Generator of a cone

- H -

Herd
Herd Equivalent
Herd Space
Hyperoval

- I -

- J -

- K -

- L -

Laguerre Plane
Linearized Polynomial

- M -

Monomial Flock

- N -

- O -

Oval
An oval in a projective plane of order n is a set of n+1 points, no three of which are collinear. Thus, an oval in a finite projective plane is an (n+1)-arc.
An oval in a Desarguesian plane of odd order is a conic (Segre).
Ovoid

- P -

Parallel Lines
Penttila-Williams Flock
Projectively Equivalent
Polarity

- Q -

Quadratic Cone
Quadric

- R -

- S -

Semi-Field Type Flock
Star Flock
Strongly Equivalent

- T -

- U -

Unital

- V -

Vertex of a Cone

- W -

- X -

- Y -

- Z -