one nonzero entry. As $\hat{J}\subset Im S_J\cap \{\mbox{ matrices of
rank } 1\}$ the following statement holds

\[Im\] $(U,V)\in \hat{J}=\pi_{34}'((\pi'_1)^{-1}(J))$ if and only if $Z=U\otimes V$ satisfies the equation $$Z_{ij;pq}^{s,s+1}=0 \mbox{ if } p-q\ge
min\{0,k_s^i-k_{s+1}^j\},\quad s=1,\dots,r-1.$$

Lemma \[kerSJ\] gives us the formula for the dimension of the kernel $$\dim \ker S_J=\sum_{s=1}^{r-1}\sum_{i,j}
min\{k^i_s,k^j_{s+1}\}.$$

We know that $GL(n,E)$ acts on $CM'_\tau$ freely. Hence if we want to estimate the dimension of the fiber of $\pi_{34}$ over $\hat{J}$ we should estimate $\dim Stab(J)-\dim ker S_J$. This difference is positive:

\[ineq\] Let $k_s\in \mathbb{N}^{d_s}$, $s=1,\dots,r$, $k^i_s\ge k^{i+1}_{s}$ then the following inequality holds $$\sum_{s=1}^r\sum_{i,j} min\{
k_s^i,k_s^j\}-\sum_{s=1}^{r-1}\sum_{i,j} min\{
k_s^i,k_{s+1}^j\}>0,$$ if there exists $s$ such that $k_s\ne 0$.

Because of the inequality $k^i_s\ge k^{i+1}_s$ we can rewrite LHS of the inequality in the form $$\begin{gathered}
\sum_{\nu=1}\left(\sum_{s=1}^r (x^{\nu}_s)^2- \sum_{s=1}^{r-1}
x^{\nu}_sx^{\nu}_{s+1}\right), \\  x^\nu_s=\#\{i\in\mathbb
N|k_s^i\ge\nu\}.\end{gathered}$$ But the first expression is a sum of positive definite quadratic forms. Thus we get the lemma.

The following statement is crucial for estimating of $\dim(\pi^{-1}_1(JNF\setminus\tilde{U}))$:

If $J$ is given by (\[J\]) with $s=1$ and $\bold{k}_1=
\bold{k}=(\vec{k}_1,\dots,\vec{k}_r)$, then $\dim
\pi_1^{-1}(J)<2n-1$ when either $r>1$ or $k_1^1>1$.

In the case $r>1$ Corollary \[Im\] implies that $\dim
\pi_{34}'((\pi'_1)^{-1}(J))\le 2n-1$. The theorem on the dimension of the fibers and 