I have a couple of questions about arguments 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:
The setup: Let $Y$ be a complex Enriques surface $Y$, so in following we work in complex setting, ie $Y$ is aspecially a compact (in modern terms proper) complex smooth surface.
LEMMA (2.6). Let $\pi: Y \to \Bbb P^1$ be a special elliptic pencil of an Enriques surface $Y$. (Ie, the general fiber of $\pi$ is smooth elliptic and special means that it has a $2$-section $R$ which is nodal curve (see Def 2.4 in the linked paper). Let $X$ be the $K3$ surface coming as double cover of $Y$ and $\sigma: X \to X$ its covering involution. Denoted by $\widetilde{\pi}: X \to \Bbb P^1$ the base change of $\pi: Y \to \Bbb P^1$. (BTW: This map is called "base change", but bc along what? Isn't this just the composition $\pi \circ p$? Or it there tacitly an involution of $\Bbb P^1$ involved?) Then there is an elliptic pencil $J(\pi): J(Y) \to \Bbb P^1$ which satisfying certain conditions (i),(ii),(iii),(iv) given in the linked paper.
I struggle with several steps in the presented proof. Could somebody look through if one or other point can be resolved:
(A): What is the argument that the inverse image $p^{-1}(R)$ of the $2$-section $R$ of $Y$ - which exists as $Y$ special - is given by a disjoint union of two nodal curves $R_+,R_-$ (here: nodal curve = smooth rational curve, ie $\Bbb P^1$; older terminology, see p 196).
Intuitivelly: Why the double cover $p:X \to Y$ resolves "doubleness" of the $2$-section $2$? In other words $R_+,R_-$ become honest sections of $X \to \Bbb P^1$.
(B): Assuming (A) has been resolved, why there exist automorphism $t: X \to X$ defined as $t(x):=r^+(\widetilde{\pi}(x)-r^-(\widetilde{\pi}(x)$ where $r^+, r^-: \Bbb P^1 \to X$ corresp to honnest sections $R_+,R_-$ from previous part. Note, that $"-"$ should come from fibrewise group structure.
Intuitivelly, this should correspond to a flip of $R_+$ and $R_-$.
I mean, there are strictly speaking two potential problems with this construction: Firstly, since $\pi: Y \to \Bbb P^1$ is elliptic pencil, only the general, but not every fibre of $\pi$ is elliptic curve - so has group structure - but does also the general fibre of $\widetilde{\pi}= \pi \circ p: X \to \Bbb P^1$ inherit group structure, recall $p:X \to Y$ is double cover?
Furthermore, assume we can manage the first problem, what about the fibres, which are not elliptic curves - so carry no group structure? There the minus $"-"$ in def. of $t$ seemingly don't make sense. Or, can one invoke there some extension result?
(C): Later one manages to construct an involution $\tau: X \to X$ and set $S =X/ \langle \tau \rangle $. Is it clear that $S$ admits elliptic pencil induced from $\widetilde{\pi}$? The thing is, $X$ is branched $2$-cover of $S$. General fibre $X_b= \widetilde{\pi}^{-1}(b)$ of $\widetilde{\pi}$ is an elliptic curve , is the quotient $S_b =X_b/ \langle \tau \rangle $ of such general fibre still elliptic curve?
(D): Next one introduces $J(\pi):J(Y) \to \Bbb P^1$ as minimal model of the pencil $S \to \Bbb P^1$ from prev. Q (C). Is it clear why the anticanonical linear system $\vert -K_{J(Y)} \vert $ must contain a fibre of $J(\pi)$? From what does it follow?
