  Let $G$ be a simple algebraic group over the complex field $\mathbb C$, $P$ a
parabolic subgroup containing $B$ its Borel subgroup, $P'$ its derived group
and $\mathfrak m$ the Lie algebra of its nilradical. The nilfibre $\mathscr N$
for this action is the zero locus of the augmentation $\mathscr I_+$ of the
semi-invariant algebra $\mathscr I=\mathbb C[\mathfrak m]^{P'}$. For $G=SL(n)$
practically nothing was known previously. The only result of comparable, but
lesser complexity, is for $\mathscr V:=\mathscr O\cap \mathfrak n$, with
$\mathscr O$ a nilptent $G$ orbit and $\mathfrak n$ the set of strictly upper
triangular matrices. Then $\mathscr V$ is equidimensional with components known
as orbital varieties, parameterised by standard tableaux whose shape is
dictated by $\mathscr O$. Here the components of $\mathscr N$ are studied for
$G=SL(n)$. They increase exponentially in $n$ with no a priori discernable
pattern. For each choice of numerical data $\mathcal C$, a semi-standard
tableau $\mathscr T^\mathcal C$, is constructed from $\mathscr T$. A
\textit{delicate and tightly interlocking} analysis constructs a set of
excluded root vectors from $\mathfrak m$ such that the complementary space
$\mathfrak u^\mathcal C$ has the following properties. First it is a subalgebra
of $\mathfrak m$. Secondly $\mathscr C:=\overline{B.\mathfrak u^\mathcal C}$
lies in $\mathscr N$ to which, thirdly, a Weierstrass section can be
associated. Fourthly $\dim \mathscr C = dim \mathfrak m-\textbf{g}$, where
\textbf{g} is the number of generators of the polynomial algebra $\mathscr I$.
Fifthly the Weierstrass section, is shown to imply that $\mathscr C$ an
irreducible component of $\mathscr N$, yet $\mathscr C$ is \textit{ only
sometimes} an orbital variety closure. The resulting Component Map $\mathscr
T^\mathcal C\mapsto\mathscr C$ is shown to be injective. Evidence for its
surjectivity is given.
