particular, the complement $V\backslash D$ is an open orbit of $G$. We call $D$ reductive if $G_D$ is so. A rational function $r\in{{\mathds C}}(V)$ is called a semi-invariant if there is a character $\chi_r:G\rightarrow {{\mathds C}}^*$ such that $g(r)=\chi_r(g)\cdot r$ for all $g\in G$. Obviously, $h$ itself is a semi-invariant.

3.  $G$ acts on $V^*$ by the dual action, with dual discriminant $D^*\subset V^*$. If $G_D$ is reductive, then $(V^*,D^*)$ is pre-homogenous. We call a linear form $f\in V^*$ generic with respect to $h$ (or simply generic, if no confusion is possible) if $f$ lies in the open orbit $V^*\backslash D^*$ of the dual action.

    There is a basis $(e_i)$ of $V$ with corresponding coordinates $(x_i)$ (called unitary) such that $G$ appears as a subgroup of $U(n)$ in these coordinates. Then $D^*=\{h^*=0\}$, where $h^*(y):=\overline{h(\overline{y})}$, $(y_i)$ being the dual coordinates of $(x_i)$.

In the sequel, we always consider linear forms which are generic with respect to $h$.

In order to study the behavior of the restriction of the linear function $f$ on the fibres $D_t:=h^{-1}(t)$, $t \neq 0$, but also on $D$ itself, the following *deformation algebra* was introduced in [@dGMS].

Let $D$ be linear free with defining equation $h$, seen 