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Linear Flocks

Given a cone C(V,S) and a line l of PG(3,q) which does not intersect C(V,S), the q planes which pass through l and do not contain the point V form a linear flock of C(V,S). We would say that this linear flock is based on the line l.

It follows immediately from Theorem 3.1[backbutton] that,

Corollary 4.1: Any cone, which admits a flock, admits a linear flock.

Flocks which are not linear are called non-linear flocks. Given the ubiquity of linear flocks, we are more interested in the existence and construction of non-linear flocks.

There are cones which admit only linear flocks. For example,

Proposition 4.1: A flock of any cone in PG(3,2) which has a flock, is linear. [proofbutton]

Also,

Proposition 4.2: If S is the complement of a line l in (i.e., an affine plane) then C(V,S) admits only linear flocks.

There are other, less trivial examples.

After we introduce coordinates and certain normalizations, the linear flocks will be easily identified algebraically.


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