Looking for a 3-fold with some property? Can any one give an example of a 3-fold X which contains an embedded ample divisor $D \cong CP^2$
with normal bundle $O(3)$ in X?
 A: Let me give an attempt of a proof of the fact that such example does not exist. 
Proof. Suppose $X$ is such a $3$-fold and let $L$ be the line bundle corresponding to the  divisor $\mathbb CP^2$. First we will prove that $Pic(X)=\mathbb Z$ and then will get a contradiction. 
Notice that $L$ has a lot of sections. In particular in a neighbourhood of $\mathbb CP^2$ for every two points $x,y$ there is section of $L$ that contains $x$ but does not contain $y$.
And also notice that for every point of $X$ there is a section that does not contain it. Hence we have a morphism $X\to P(H^0(L)^*)$. This morphism can not contract anything since $L$ is ample. The map is an embedding on the neighbourhood of $D$ and so the image can have singularities at most in codimension $3$.
From this it should follow (I guess), that we can apply Lefshetz that says $Pic (X)=Pic(D)=\mathbb Z$. So $X$ is a Fano with $Pic=\mathbb Z$. Torsten Ekedahl explained how now one can deduce contrudiction, see his comment.
A: There exists such variety which is singular with Gorenstien singularities.
Let $S$ be a del Pezzo surface of degree $d$ and let ${\mathcal L} = O_S(−K_S)$. 
Consider the $\mathbb P^1$ -bundle 
$\mathbb{P} = \mathbb{P}_S (\mathcal{O}_S \bigoplus \mathcal{L})$. 
Now the variety $X$ can be constructed as a contruction of a zero divisor. The map $\mathbb{P}\to X$ given by the linear system 
${\mathcal{O}}_{\mathbb{P}}(n), \quad n ≫ 0.$ 
It contracts the negative section. Since $−K_{\mathbb P} ∼ O_{\mathbb P}(2),$ 
the variety $X$ is a Fano threefold of index $2$ and degree $8d$ with canonical Gorenstein singularities. For $S = {\mathbb P}_2$ we have $−K^3_X = 72$ and 
$X ≃ \mathbb{P}(3, 1, 1, 1)$ is a weighted projective space.
