Igusa uses Weil's language, in a modified/enhanced version that deals
with reduction mod primes. (My memory is that there is a paper of Shimura from the 50s that develops this language.) It's not so easy to read it carefully, unfortunately.
Chow's method for constructing Jacobians (explained in his paper in the American Journal
from the 50s, again if memory serves) is, I think, as follows:
take $Sym^d C$ for $d > 2g - 2$. The fibres of the map $Sym^d C to Pic^d(C)$
are then projective space of uniform dimension (by Riemann--Roch), and so it is not so hard to quotient out
by all of them to construct $Pic^d(C)$ (for $d > 2g - 2$), and hence to construct the Jacobian. (I hope that I'm remembering correctly here; if not, hopefully someone will correct me.)
I think that this should be contrasted with the more traditional method of considering $Sym^g C$, which maps birationally to $Pic^g(C)$, i.e. with fibres that are generically points, but which has various exceptional fibres of varying dimensions, making it harder to form the quotient, thus inspiring in part Weil's "group chunk" method where he uses the group action
to form the quotient (in an indirect sort of way), and consequently loses some control of the
situation (e.g. he can't show that the Jacobian so constructed is projective). I should
also say that it's been a long time since I looked at this old 1950s literature, and I'm not
completely confident that I understand its thrust (i.e. I'm not sure what was considered easy and what was considered hard, and what was considered new and innovative in various papers as contrasted to what was considered routine), so take this as a very rough guide only.
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