I am trying to read this paper http://arxiv.org/pdf/math/9902092.pdf.
I have some trouble at the end in Corollary 3.27, and also Proposition 3.24.
0)Morally speaking my main trouble is that i don't quite follow how in that paper they actually prove the fact that in elliptic K3 surfaces there are infinitely many rational nt multisection. More specifically i have the following questions(3) is the question that leave me really puzzled, because i see that i'm not getting the main point of the paper, that is sketched in the three words "by deformation theory" at the end)
1)When they speak(Corollary 3.27) of "Dividing the cocycle defining S' by different primes", i don't really understand what they are referring to(the cocycle defining the elliptic fibration?), and so i don't understand the relation between the K3 surface obtained by such a cocycle and the original one.
2)In Lemma 3.24 they use the notation ${\epsilon'}^{d_{\epsilon}}$, i don't know what they are referring to.
3)And the most troubling for me:In Corollary 3.27, they say that the $\epsilon_p$ obtained by the procedure of dividing the defining cocycle by p, has "by deformation theory a rational multisection of degree divisible by $d_{\epsilon_p}$". It seems that the very generic "by deformation theory" is refferring to Lemma 2.12,and the previous Proposition 2.2. But in both cases it is on another deformed surface that i got a rational curve , so i don't see why we get one on $\epsilon_p$, and then i don't see how you get one in the original surface, but probably this is explained by the answer to 1) and 2))
I thank a lot in advance anyone who can help me understand these questions.