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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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Finite extension of K(x) with extra structure: definable over field of invariants?

Let $K$ be an algebraically closed field, and let $\sigma$ be an automorphism on $K$. Set $k=K^\sigma$. Consider the rational function field $K(x)$ and extend $\sigma$ to $K(x)$ by $\sigma(x)=x$, henc …