Is there anyone here who has worked through Mumford's Tata Lectures on Theta II and can help me with my difficulties in following what seems to be confusing and inconsistent notation?
I have had more than one confusion in trying to follow the expressions in the main results of the book, but my main issue at the moment involves Mumford's notation for various (translations of) theta divisors $\Theta \subset \mathrm{Jac}(C)$. Namely, early on Mumford defines $\Theta$ as the locus of divisor classes represented by divisors of the form $\sum_{i = 1}^{g - 1} P_i - (g - 1) \cdot (\infty)$ and uses this to construct variables and polynomial relations to define $\mathrm{Jac}(C) \setminus \Theta$ as an affine subset of $\mathrm{Jac}(C)$. He sticks with this definition of $\Theta$ later, for example on page 3.98. However, at the beginning of section 10 (page 3.155), $\Theta$ is instead a translation of this by a 2-torsion point depending on the choice of some $(g + 1)$-element subset $U$ of the branch points. The subsequent characterization of $\vartheta$ on the same page (that it vanishes to order 1 on $\Theta$) is then compatible (I think) with previous formulas for the theta function $\vartheta$. Yet in the very same section, two pages later to state Proposition 10.1, he again uses the construction of the affine subset $\mathrm{Jac}(C) \setminus \Theta$ which used the original definition of $\Theta$.
This may sound like a very superficial inconsistency, but I'm having serious trouble working out how all of the results are supposed to fit together without resolving the issue of fixing a particular translation of the theta divisor. Any clarification from someone who has a deeper familiarity with this part of the classical theory would be greatly appreciated.
