Help to understand the proof of the Riemann Mumford relation

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Here i post a file where from page 617 to 618 there is the proof of the Riemann mumford relation that is the theorem 1.13.

My problem is to understand the beginning of that proof. In particular why the pull-back line bundle is invariant under the involution sheet exchange and then How does the author compute the dimension of the space $h^0(C^{'},\tilde{L})$?
Any suggestions?

Joe Harris's Article