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 $2$?



An example of simultaneous Du Val resolution
Let's start with the second set of questions, because that partially explains what happens in the first.


**Why [is] the singularity at the vertex of $C\_V$ the same as $\mathbb C^3/(x∼−x)$?**


Well, the Veronese is the image of the map taking the degree $2$ monomials in $3$ variables, so its homogenous coordinate ring is the ring $\mathbb C[x^2,y^2,z^2,xy,xz,yz]$, which is exactly the invariant subring under the action $(x,y,z)\sim (-x,-y,-z)$, so the cone $C\_V$, which is just $\mathrm{Spec}\ \mathbb C[x^2,y^2,z^2,xy,xz,yz]$, will be the quotient of the affine space by that action.


**And [why does] the modification the exceptional locus ha[ve] pure dimension $2$?**


This is true for all $\mathbb Q$-factorial singularities. See [this answer](https://mathoverflow.net/a/45193/10076) to another MO question. 


So let's get to the first set of questions: 


**How [can one] describe the blowing up $C\_F$ along $C\_l$[?]**


The funny thing is that what makes this work is that the singularity of $C\_F$ is *worse* than that of $C\_V$ in the sense that it is not $\mathbb Q$-factorial (this is because the Picard group of $F$ has rank $2$ as opposed to the rank of the Picard group of $V$ being $1$). Because of that it admits a *small morphism* (see the linked answer for details). The non-$\mathbb Q$-factoriality means that there is a Weil divisor which is not $\mathbb Q$-factorial. In this case that's $C\_l$, so blowing it up you get a small morphism. 


**[W]hy does the special fibre $C\_4$, after blowing up, have only Du Val singularities?** and **What is the strict transform of $C\_4$?**


The above allows for the fibers to remain normal and shows that the strict transform=preimage of $C\_4$ is just the blow up of $C\_4$ at its vertex which is obviously Du~Val and nothing else changes, because $C\_l$ is Cartier everywhere else.


