Let $X$ be a $K3$ surface and $f: X \to X$ a non-symplectic morphism (ie 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.
Question: How to see that the quotient $Y:= X/ \langle f \rangle$ (which exists as $X$ projective) is either a rational surface or birational to an Enriques surface and when is it possible to distinguish these two cases?
This is claimed in Huybrecht's Lecture on K3 without proof in Lemma 15.4.8 (p 330).
There the is a well known Enriques classification of surfaces in terms of cohomological invariants and $f$ induces clearly comparison maps between those of $X$ and $X/ \langle f \rangle$, but do we know a priori sumething about these maps? (surj, inj, etc)
The question is how the invariants of $X$ control those of $X/ \langle f \rangle$ trough this map?
Note that Kodaira dimension is a birational invariant, and generically our map $X \to Y$ is an étale $n$- cover, and on étale covers the plurigenera not change, as for étale map of Gorenstein (...in order to keep canonical sheaf invertible) surfaces $e: X \to Y$ we have $\omega_X= e^*\omega_Y$, so this raises the question if there is a formula relating plurigenera $P_n$ of $X$ and $Y$, as plurigenera as birational invariants not change with respect to ramified covers.
So the key question: If $e: X \to Y$ is an finite, etale cover of normal Gorenstein surfaces, is there some relation between plurigenera $P_n$ of $X$ and $Y$?
This is motivated by argumenation in the proof of Lemma 2.6 (see absol page 199, rel p 9) from Shigeyuki Kondo's paper Enriques surfaces with finite automorphism groups:
There one constructed certain involution $\tau$ on K3 surface $X$ $2$-covering an Enriques surface $Y$, and the question is why $S:=X/ \langle f \rangle$ is rational. (in order then to proceed to pick it's minimal model).
Compare with this question.
