\le 0$ for all $s$ in a small neighbourhood of $0$.

Our definition of [[$\mathcal F$[[ISI]{}]{}]{}]{} deformation contains the condition that $f_0$ is [[$\mathcal F$[[ISI]{}]{}]{}]{} and it implies that $f_s$ is [[$\mathcal F$[[ISI]{}]{}]{}]{} for any $s$ in some small neighbourhood of $0$.

A constant degree deformation such that, for all $s$, $f_s$ is a polynomial in 2 variables with isolated singularities in the affine, is a [[$\mathcal F$[[ISI]{}]{}]{}]{} deformation.

We attach to the family $P$ the following hypersurface: $${{\mathbb Y}}= \{ ([x:x_0], s, t) \in {{\mathbb P}}^n \times {{\mathbb C}}\times {{\mathbb C}}\mid \tilde 
P(x,x_0,s) - tx_0^d = 0 \},$$ where $\tilde P$ denotes the homogenised of $P$ by the variable $x_0$, considering $s$ as parameter varying in a small neighbourhood of $0\in {{\mathbb C}}$. Let $\tau : {{\mathbb Y}}\to {{\mathbb C}}$ be the projection to the $t$-coordinate. This extends the map $P$ to a proper one in the sense that ${{\mathbb C}}^n\times {{\mathbb C}}$ is embedded into ${{\mathbb Y}}$ and that $\tau_{|{{\mathbb C}}^n\times {{\mathbb C}}} = P$. Let $\sigma : {{\mathbb Y}}\to 
{{\mathbb C}}$ denote the projection to the $s$-coordinates. We shall use the notations ${{\mathbb Y}}_{s,{{\mathbb C}}} := {{\mathbb Y}}\cap \sigma^{-1}(s)$, ${{\mathbb Y}}_{{{\mathbb C}}, t} := {{\mathbb Y}}\cap 
\tau^{-1}(t)$ 