I was wondering whether or not there is some kind of theory of algebraic geometry over the field of Surreal and Surrcomplex numbers?

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    $\begingroup$ I believe that there is some theorem to the effect that “algebraic geometry over any algebraically closed field of characteristic 0 may as well be over $\mathbb{C}$” which would include the surcomplex numbers, I’m not sure about the surreals though. $\endgroup$
    – Alec Rhea
    Jan 18, 2019 at 1:09
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    $\begingroup$ @AlecRhea I think there's some theorem of the sort also for real closed fields, saying you might as well do it over the reals. $\endgroup$ Jan 18, 2019 at 9:28
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    $\begingroup$ @DenisNardin Ah, that seems reasonable, if either of us could produce a reference it'd be ideal though :-). I think it's worth mentioning that these theorems aren't necessarily telling us that there is no interesting and new 'geometry' that takes place over the surreals/surcomplex numbers which can be captured algebraically, but rather that the standard tools of modern algebraic geometry will not be sensitive to any of the new phenomenon we might encounter. It also seems like model-theoretic arguments are proving these results (ref?), so the tools under discussion would be first order (cont.) $\endgroup$
    – Alec Rhea
    Jan 25, 2019 at 23:46
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    $\begingroup$ and perhaps some second order machinery would be more successful at algebraically differentiating between geometry over the real/complex vs surreal/surcomplex numbers. $\endgroup$
    – Alec Rhea
    Jan 25, 2019 at 23:47


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