Let $V\subseteq\mathbb{P}^5$ be the Veronese surface and $C_V\subseteq\mathbb{P}^6$ the cone over it. Let $F\subseteq\mathbb{P}^5$ be the scorll $\mathrm{Proj}_{\mathbb{P}^1}(\mathcal{O}(2)+\mathcal{O}(2))$ embedded with the relative $\mathcal{O}(1)$, and $C_F$ the cone over it. If we cut $C_V$(resp. $C_F$) with a pencil of hyperplanes, then the hyperplane through the vertex gives a cone $C_4\subseteq\mathbb{P}^5$ over the rational normal curve of degree $4$; the general member will be isomorphic to $V$(resp. $F$). The family $C_F$ admits a simultaneous Du Val resolution: blow up $C_l$ cone over a line on $F$. The family obtained from $C_V$ does not admits a simultaneous Du Val resolution: the singularity at the vertex is $\mathbb{C}^3/(x\sim -x)$; hence any birational modification the exceptional locus must be of pure dimension two, therefor any modification of $C_V$ would introduce a new component to the central fibre. An other reason is that $K_{\bar{C_4}}^2=8$ and $K_{\bar{V}}^2=9$, a necessary condition of a family $X\longrightarrow B$ admit a simultaneous Du Val resolution is the $K_{\bar{X}_b}^2$ is locally constant for $b\in B$, where $\bar{X}$ is denoted as the minimal resolution of $X$.

My questions are the following:

[1] How to describe the blowing up $C_F$ along $C_l$, why the special fibre $C_4$ after blowing up has only Du Val singularities? What is the strict transform of $C_4$?

[2] Why the singularity at the vertex of $C_V$ is $\mathbb{C}^3/(x\sim -x)$? And the modification the exceptional locus has pure dimension $2$?