  Let $\Lambda$ be a local truncated path algebra over an algebraically closed
field $K$, i.e., $\Lambda$ is a quotient of a path algebra $KQ$ by the paths of
length $L+1$, where $Q$ is the quiver with a single vertex and a finite number
of loops and $L$ is a positive integer. For any $d>0$, we determine the
irreducible components of the varieties that parametrize the $d$-dimensional
representations of $\Lambda$, namely, the components of the classical affine
variety ${\rm\bf{Rep}}_{d}(\Lambda)$ and -- equivalently -- those of the
projective parametrizing variety ${\rm GRASS}_d(\Lambda)$. Our method is to
corner the components by way of a twin pair of upper semicontinuous maps from
${\rm\bf{Rep}}_{d}(\Lambda)$ to a poset consisting of sequences of semisimple
modules.
  An excerpt of the main result is as follows. Given a sequence ${\bf S} =
({\bf S}_0, ..., {\bf S}_L)$ of semisimple modules with $\dim \bigoplus_{0 \le
l \le L} {\bf S}_l = d$, let ${\rm\bf{Rep}}\, {\bf S}$ be the subvariety of
${\rm\bf{Rep}}_{d}(\Lambda)$ consisting of the points that parametrize the
modules with radical layering ${\bf S}$. (The radical layering of a
$\Lambda$-module $M$ is the sequence $\bigl(J^l M / J^{l+1} M\bigr)_{0 \le l
\le L}$, where $J$ is the Jacobson radical of $\Lambda$.) Suppose the quiver
$Q$ has $r \ge 2$ loops. If $d \le L+1$, the variety
${\rm\bf{Rep}}_{d}(\Lambda)$ is irreducible. If, on the other hand, $d > L+1$,
then the irreducible components of ${\rm\bf{Rep}}_{d}(\Lambda)$ are the
closures of the subvarieties ${\rm\bf{Rep}}\, {\bf S}$ for those sequences
${\bf S}$ which satisfy the inequalities $\dim {\bf S}_l \le r \dim {\bf
S}_{l+1}$ and $\dim {\bf S}_{l+1} \le r \dim {\bf S}_l$ for $0 \le l < L$. As a
byproduct, the main result provides generic information on the modules
corresponding to the irreducible components of the parametrizing varieties.
