as a morphism $h:V\rightarrow T:=\operatorname{\textup{Spec}\,}{{\mathds C}}[t]$.

1.  Let $E\in \Theta_V(-\log\,D)$ be the Euler field $E=\sum_{i=1}^n x_i\partial_{x_i}$. Call $$\Theta_{V/T}(-\log\,D):=\{\vartheta\in\Theta_V(-\log\,D)\,|\,\vartheta(h)=0\}$$ the module of relative logarithmic vector fields. $\Theta_{V/T}(-\log\,D)$ is ${\mathcal{O}}_V$-free of rank $n-1$, and we have a decomposition $\Theta_V(-\log\,D)={\mathcal{O}}_V E \oplus \Theta_{V/T}(-\log\,D)$.

2.  The ideal $J_h(f):=df(\Theta_{V/T}(-\log\,D)) \subset {\mathcal{O}}_V$ is called the Jacobian ideal of the pair $(f,h)$. The quotient ${\mathcal{O}}_V/J_h(f)$ is the Jacobian algebra (or deformation algebra) of $(f,h)$.

Notice that $\Theta_{V/T}(-\log\,D)$ was called $\Theta_V(-\log\,h)$ in [@dGMS]. It was shown in loc.cit., section 3.2, that if $f$ is generic with respect to $h$, then $h_*{\mathcal{O}}_V/J_h(f)$ is ${\mathcal{O}}_T$-free of rank $n$, and generated by $(f^i)$ for $i=0,\ldots,n-1$. Moreover, it is interpreted as the relative tangent space $T^1_{{\mathcal{R}}_h/{{\mathds C}}}(f)$ of the deformation theory of $f$ with respect to the group ${\mathcal{R}}_h$ of right-equivalences preserving all fibres of $h$.

Denote by $(\Omega^\bullet_{V/T}(\log\,D),d):=(\Omega^\bullet_V(\log\,D)/(\Omega^{\bullet-1}_V(\log\,D)\wedge h^*\Omega^1(\log\,\{0\}),d)$ the relative logarithmic de Rham complex of $h$ as studied, under the name $\Omega^\bullet(\log\,h)$ in [@dGMS section 2.2]). This relative logarithmic complex is used in the definition of the family of Gauß-Manin-systems resp. Brieskorn lattices in loc.cit.

Let $h$ and $f$ as above. Define $$\label{eq:GM-Brieskorn}
\begin{array}{rclcrcl}
G(\log\,D) & := & \frac{H^0(V,\Omega^{n-1}_{V/T}(\log\,D)[\theta,\theta^{-1}])}{(\theta d-df\wedge)H^0(V,\Omega^{n-2}_{V/T}(\log \,D)[\theta,\theta^{-1}])}
&;&
G(*D) & := & \frac{H^0(V,\Omega^{n-1}_{V/T}(*D)[\theta,\theta^{-1}])}{(\theta d-df\wedge)H^0(V,\Omega^{n-2}_{V/T}(*D)[\theta,\theta^{-1}])}
\\ \\
