Let $x\_1^4+x\_2^4+x\_3^4+x\_4^4=0 \subset \mathbb{P}^4$ be the Fermat K3 surface. Is it possible to start from some involution on $\mathbb{P}^3$, do blow-ups to get rid of fixed points and then quotient with respect to resulting involution to get an Enrique surface inside the quotient of some blow-up of $\mathbb{P}^3$.


Or


can you give an example of an Enrique surface inside a simply-connected 3-fold?



How to construct Enriques surface from Fermat K3
Let me show that the answer to your last question is **yes**, by explaining how to construct an Enriques surface inside a simply connected threefold. 


Let $\psi \colon V \to \mathbb{P}^3$ be a double covering branched on a smooth quartic surface and let $\tau \in \textrm{Aut}(V)$ be an involution fixing eight points. Set $X=V/ \tau$. Then one proves that $X$ is a *rational* threefold with eight singular points of type $\frac{1}{2}(1,1,1)$. The anti-canonical divisor $-K\_X$ is not Cartier but we have 
$-K\_X \equiv\_{Q} H$, where $H$ is an ample divisor on $X$ with $H^3=8$ such that the general element of $|H|$ is a smooth Enriques surface.


Now consider any desingularization $\pi \colon Y \to X$. Since $X$ is rational, $Y$ is a smooth rational variety, hence $\pi\_1(Y)=0$. Moreover $X$ has only isolated singularities, so the general element of $|\pi^\*H|$ is again a smooth Enriques surface.


**Example.** Let $V \subset \mathbb{P}(1^3, \; 2)$ defined by the equation $$u^2=x^4+y^4+z^4+t^4,$$
where $x,\; y, \; z,\; t$ have weight $1$ and $u$ has weight $2$, and consider the involution $\tau \colon \mathbb{P}(1^3, \; 2) \to \mathbb{P}(1^3, \;2)$ given by $$\tau(x:y:z:t:u)=(x:y:-z:-t:-u).$$
Then $V$ is naturally a double cover of $\mathbb{P}^3$ branched on a Fermat quartic surface and moreover $V$ is $\tau$-invariant. The restricted involution $\tau|\_V$ fixes the eight points of $V$ given by $u=z=t=0$ and $u=x=y=0$.


For further details see [I. Cheltsov, Rationality of Enriques-Fano threefold of genus five, Isvestiya Math. **68** (2004)](http://www.maths.ed.ac.uk/cheltsov/pdf/izv04b.pdf) and the references given therein. 


