An example of simultaneous Du Val resolution

Question

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 $2$?

Answer 1

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 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.