and ${{\mathbb Y}}_{s,t} := {{\mathbb Y}}_{s,{{\mathbb C}}}\cap \tau^{-1}(t) = {{\mathbb Y}}_{{{\mathbb C}}, t}\cap 
\sigma^{-1}(s)$. Note that ${{\mathbb Y}}_{s,t}$ is the closure in ${{\mathbb P}}^n$ of the affine hypersurface $f_s^{-1}(t)\subset {{\mathbb C}}^n$. Let ${{\mathbb Y}}^{{\infty}}:= {{\mathbb Y}}\cap \{ x_0=0\} = \{ 
P_d(x,s) = 0\}\times {{\mathbb C}}$ be the hyperplane at infinity of ${{\mathbb Y}}$.

The singular locus of ${{\mathbb Y}}$, namely: $$\label{eq:singY}
 {{\rm{Sing\hspace{2pt}}}}{{\mathbb Y}}= \{ x_0 =0, \frac{\partial P_d}{\partial x}(x,s) =0, \  P_{d-1}(x,s)=0, 
\frac{\partial P_d}{\partial s}(x,s)=0 \} \times {{\mathbb C}}$$ is a subset of ${{\mathbb Y}}^{{\infty}}$ and is a product-space by the $t$-coordinate. It depends only on the degrees $d$ and $d-1$ parts of $P$ with respect to the variables $x$.

For a fixed $s$ in a small neighbourhood of $0\in {{\mathbb C}}$, the singular locus of ${{\mathbb Y}}_{s,{{\mathbb C}}}$ is the analytic set:

$$\label{eq:sing}
{{\rm{Sing\hspace{2pt}}}}{{\mathbb Y}}_{s,{{\mathbb C}}} = \Sigma_s \times {{\mathbb C}}\subset {{\mathbb Y}}_{s,{{\mathbb C}}}\cap {{\mathbb Y}}^{{\infty}}.$$

It is the union, over all $t\in {{\mathbb C}}$, of the singularities of the fibers ${{\mathbb Y}}_{s,t}$ situated on the part at infinity ${{\mathbb Y}}_{s,t}^{{\infty}}:= {{\mathbb Y}}_{s,t} \cap H^{{\infty}}$. Besides these singularities, there might be points $(p,s,t)\in {{\mathbb Y}}^{{\infty}}$ where the fiber ${{\mathbb Y}}_{s,t}$ is not singular but it is tangent to the 