(The title is mostly a joke, because Eckardt points really weren't
the main difficulty.) Well, after another bunch of hours rewriting
this proof in all aspects, I seem to have come to a final-so-far
version of it: the statement is that weak approximation holds in
places of good reduction for smooth cubic surfaces over the function
field of a curve over an algebraically closed field. If that's
Chinese to you, here's a slightly more understandable version of the
same: consider a homogeneous cubic equation in four unknowns (say,
`X`, `Y`, `Z` and `T`) whose
coefficients are rational functions of one indeterminate (say,
`λ`) having themselves complex coefficients: something
like `c`_{3000}(`λ`)·`X`³ +
`c`_{2100}(`λ`)·`X`²·`Y`
+
`c`_{2010}(`λ`)·`X`²·`Z`
+ … +
`c`_{0003}(`λ`)·`T`³ = 0
where the
`c`_{ijkl}
are rational functions in `λ` with complex
coefficients, and we are trying to solve for the variables
`X`, `Y`, `Z` and `T` also as
rational functions in `λ`; well, the idea is that if
you fix a finite number of `λ` and for each one choose
a solution of the corresponding equation (when specialized to that
value), then there should be a parametric solution (one where
`X`, `Y`, `Z` and `T` are rational
functions in `λ`; I insist, *rational*) which
interpolates them all, and actually, in a way I won't try to describe,
one can fix the derivatives to any order of the variables
`X`, `Y`, `Z` and `T`, not just
their values at certain points; now actually this doesn't work for all
`λ`: there are a certain finite number of them which
can't be handled (places of “bad reduction”), essentially
those in which the specialized equation becomes too degenerate in
certain ways.

Now that I've finished that, I can perhaps start working on the piles and piles of emails that I still have to reply to.