  We consider the reconstruction of spike train signals of the form $$F(x) =
\sum_{i=1}^d a_i \delta(x-x_i),$$ from their moments measurements $m_k(F)=\int
x^k F(x) dx = \sum_{i=1}^d a_ix^k$. When some of the nodes $x_i$ near collide
the inversion becomes unstable. Given noisy moments measurements, a typical
consequence is that reconstruction algorithms estimate the signal $F$ with a
signal having fewer nodes, $\tilde{F}$. We derive lower bounds for the moments
difference between a signal $F$ with $d$ nodes and a signal $\tilde{F}$ with
strictly less nodes, $l$. Next we consider the geometry of the non generic case
of $d$ nodes signals $F$, for which there exists an $l<d$ nodes signal
$\tilde{F}$, with moments \begin{align*}
m_0(\tilde{F})=m_{0}(F),\ldots,m_{p}(\tilde{F})=m_{p}(F),&& p>2l-1 .
\end{align*} We give a complete description for the case of a general $d$,
$l=1$ and $p=2$. We give a reference for the case $p=2l-1$ which can be
inferred from earlier work.
