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Here we recall some basic notions of classical projective geometry
(e.g. see [Bert1], [Bert2] and [SK]).
A linear complex 
of 
is by definition the system of
all lines of whose plückerian coordinates satisfy a given
homogeneous linear equation 
.
A linear complex consisting of all the lines through a given
line (the axis of ) is called special.
Given a symplectic form on 
, it is readily seen that the isotropic
lines of the associated projective space (corresponding to the maximal
totally isotropic subspaces of 
) form a non-special linear complex, and
conversely any non-special linear complex in can be thought of as the
set of isotropic lines with respect to a symplectic form. (It thus follows that
the full stabilizer in 
of a given non-special linear complex is
isomorphic to the projective general symplectic group 
).
A linear congruence of is the set 
of all lines that are common
to two given linear complexes 
, 
.
Let 
(where 
and 
are the
equations of and respectively) be the pencil of complexes generated
by
and . If all complexes in 
are special, then ``degenerates" into
a plane and the lines through the intersection of the axes of and .
If no complex in is special, is said to be elliptic. In this
case, there are
exactly two complexes in which are special when viewed in
,
being a quadratic extension of 
, and their axes are skew lines
over , called the ``imaginary" axes of .
Otherwise, either contains exactly two special complexes, w.l.o.g. say
, , or contains a unique special complex . In the former
case is called hyperbolic, and consists of the lines leaning on
the axes of and . In the latter case is called parabolic and
consists of lines through the axis of .
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