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Are there any interesting algebro-geometric phenomena that happen over large algebraically closed fields of characteristic 0 and do not happen over $\mathbb{C}$ ("large" means cardinality larger than continuum)? I won't define interesting here, let's say something that can be appreciated by a person who doesn't know set theory.

A precise version of Lefschetz principle was mentioned on MO e.g. here but I do not see if it excludes such thing from happening.

アニメ愛好家も人です
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    When I've read the title I've remembered an easy proof of Hilbert's Nullstellensatz which only works over uncountable fields - most (all?) proofs easily reduce to showing that a field $K$ finitely generated as an algebra over another field $k$ is an algebraic extension. It is easy to see $K/k$ is countably-dimensional. If $k$ is uncountable and $x\in K$ is transcendental over $k$, then $(x-\alpha)^{-1},\alpha\in k$ would be an uncountable linearly independent set. Of course, this one works over $\mathbb C$, so it doesn't answer the question. – Wojowu Jun 19 '19 at 11:46
  • https://meta.mathoverflow.net/questions/4200/flood-of-similar-new-users – Squid with Black Bean Sauce Jun 19 '19 at 15:05

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