Let $X$ be an algebraic K3 surface (for sake of simplicity, with base field of char $\neq 2$) and $f: X \to X$ a non-symplectic morphism (i.e. non-symplectic in sense of that that the induced action on $H(X,K_X)=H^0(X, \Omega_X^2)$ is not trivial) of finite order say $n$.
We consider the quotient $Y:=X/ \langle f \rangle$.
On the other hand there are cyclic covers, as explained in Lazarsfeld's Positivity in algebraic geometry I in Remark 4.1.7 constructed roughly from "downside" by taking a $n$-torsion line bundle $L$ and a section on it $s \in L(Y)$, then this gives rise to a ramified $n$-cover $r:X \to Y$ branched along $V(s)$.
(Firstly, by the way this construction behaved nicely with respect forming dualizing sheaf of $\omega_X$; there are explicit formulae for $\omega_X$ (and $r_* O_X$) in terms of $L$ and $\omega_Y$. Does any body know a reference proving these rigorously?)
Question: How "close" are these cyclic cover constructions to "cyclic quotients" described above $Y:=X/ \langle f \rangle$ induced by cyclic non-symplectic automorphisms?
Obviously, the direction from a cyclic cover to such one is clear, but what about other direction? Can we find find to any such quotient $Y:=X/ \langle f \rangle$ a cyclic cover a la Lazarsfeld in "appropriate" sense?
By "in appropriate sense" I meant that a priori $Y=X/ \langle f \rangle$, may be singular, where cyclic covers have by construction smooth base. But what if we firstly resolve singularities of $Y$ and the consider the pullback of $X$ along resolution map?
Motivation: In Huybrecht's Lecture on K3 is claimed without proof in Lemma 15.4.8 (p 330) that the quotient $X/ \langle f \rangle$ (which exists as $X$ projective) is either a rational surface or birational to an Enriques surface.
The argument shall base on comparing birational invariants of Enriques classification on minimal surfaces. But in order to do this we need a tool(s) how to calculate them knowing those of $X$ :)
Here I asked a separate question about this problem generally, but it seems to be not immediate clear how to compare birational invariants determining classification types of $X$ and $Y$.
But for cyclic covers it is easier to handle, so that's my motivation, if its possible to reduce the problem to cyclic covers.
