geometry ag.algebraica: Is this proof of the wrong Appell-Humbert theorem?

In the book (very beautiful) "Diophantic Geometry" of Hindry and Silverman there is a proof of the Appell-Humbert theorem in Exercise A.5.5. I think it contains a serious error and I want to know if I did it right.

The declaration of the Appell-Humbert theorem [e.g. ] is that there is a bijection between the packets of lines in a complex bull and pairs $ (H, α) $ where $ H $ It's a certain Hermitian way and $ alpha $ It is a semi-character in the network. In the proof given in the book such a pair is built for a divider using the corresponding theta function. The problem is that, in a general complex bull, not all packets of lines come from a divisor (you can find an example in [J.D.Lewis, A survey of the Hodge Conjecture, Lecture 5]).