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For a few days, I have been working on the following problem, from Qing Liu's book:

Let $\mathcal{O_K}$ be a discrete valuation ring with uniformizing parameter t and residue characteristic $\neq 2,3$. Let $X = Proj \mathcal{O}_K[u,v,w] /(u^2w+v^3+t^6w^3)$. Determine the minimal desingularization of X.

I have been able to desingularize X by blowing it up first at $(t,u,v)$ to obtain $\widetilde{X}$and then, calling the additional coordinates in the blow-up by $(x_1,x_2,x_3)$, blowing $\widetilde{X}$ up at $(t,x_2,x_3)$ and then normalizing the corresponding scheme one gets. Let us call this scheme, obtained from two blow-ups and normalizing for $Z$, and the scheme we get from blowing up twice for $Y$. I then got that: $$Z = Proj \widetilde{X}[v_1,v_2,v_3]/(v_1x_2-tv_2,v_1x_3-tv_3,v_2x_3-v_3x_2,v_1v_2^2+v_3^3+v_1^3)$$ and $$\widetilde{X} = Proj X[x_1,x_2,x_3]/(x_1u-tx_2,x_1v-tx_3w,x_2^2w+x_3^2v+t^4x_1^2).$$ However, I might have made a mistake, but I believe Z to be regular. Now, one can check that there is an exceptional E divisor in $Z_k$, in $Z$ has defining equations $$(t,u,v,v_1,v_3).$$

I want to contract this E, so I should find an ample effective cartier divisor D such that $\mathcal{O}_Z(D)$ is generated by global sections, $\mathcal{O}_Z(D)_{Z_K}$ (restriction to the generic fiber) is ample, $Supp D \cap E = \emptyset$ and for any other curve on $Z$, the restriction of $\mathcal{O}_Z(D)$ to it is ample.

Now, I get that we should look for such a divisor in the generic fiber, and I believe that I might have found a candidate, namely $D=(v_3+v_1,v_2)$. I also believe that by Riemann-Rochlike theorems, $3D$ should be ample and generated by global sections.

However, here is my problem:

I want to see, in concrete, how I can calculate the contraction morphism AND the resulting scheme one gets. I just don't want to wave my hands and say that we contract, I would like to do one calculation in all detail, just to know how it works. But when I have been trying to do this, I get hopefully stuck, and I end up getting nowhere. Since my computational powers fail me, I would be more than thankful for someone taking their time and actually showing me how this calculation is done.

Thank you for your time.

Update: I believe that the normalization can be wrong, but I am not sure.

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