**Sections**

## 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 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.

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.

**Sections**