  We consider the category $\mathcal S(n)$ of all pairs $X = (U,V)$, where $V$
is a finite-dimensional vector space with a nilpotent operator $T$ with $T^n =
0$, and $U$ is a subspace of $V$ such that $T(U) \subseteq U$. Our main
interest in an object $X=(U,V)$ are the three numbers $uX=\dim U$ (for the
subspace), $wX=\dim V/U$ (for the factor) and $bX=\dim {\rm Ker} T$ (for the
operator). Actually, instead of looking at the reference space $\Bbb R^3$ with
the triples $(uX,wX,bX)$, we will focus the attention to the corresponding
projective space $\Bbb T(n)$ which contains for a non-zero object $X$ the
level-colevel pair {\bf pr}$X = (uX/bX,wX/bX)$ supporting the object $X$.
  We use $\Bbb T(n)$ to visualize part of the categorical structure of
$\mathcal S(n)$: The action of the duality $D$ and the square $\tau_n^2$ of the
Auslander-Reiten translation are represented on $\Bbb T(n)$ by a reflection and
a rotation by $120^\circ$ degrees, respectively. Moreover for $n\geq 6$, each
component of the Auslander-Reiten quiver of $\mathcal S(n)$ has support either
contained in the center of $\Bbb T(n)$ or with the center as its only
accumulation point.
  We show that the only indecomposable objects $X$ in $\mathcal S(n)$ with
support having boundary distance smaller than 1 are objects with $bX=1$ which
lie on the boundary, whereas any rational vector in $\Bbb T(n)$ with boundary
distance at least 2 supports infinitely many indecomposable objects. At
present, it is not clear at all what happens for vectors with boundary distance
between 1 and 2. The use of $\Bbb T(n)$ provides even in the (quite
well-understood) case $n = 6$ some surprises: In particular, we will show that
any indecomposable object in $\mathcal S(6)$ lies on one of 12 central lines in
$\Bbb T(6)$.
  The paper is essentially self-contained, all prerequisites which are needed
are outlined in detail.
