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acknowledged an answer in a particular case and made clear I am interested in the general case
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Jérôme Poineau
Jérôme Poineau
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Let $X$ be a smooth projective curve over an algebraically closed field $k$. Let $S$ and $T$ be two disjoint infinite subsets of $X(k)$. Does there exist a morphism $f:X \to \mathbb{P}^1$ and two infinite subsets $A$ and $B$ of $\mathbb{P}^1(k)$ such that the preimages of $A$ and $B$ by $f$ lie inside $S$ and $T$ respectively?

Edit: As pointed out by Dragon, this may fail when $k$ is countable, but I would like to know the answer in the general case.

Let $X$ be a smooth projective curve over an algebraically closed field $k$. Let $S$ and $T$ be two disjoint infinite subsets of $X(k)$. Does there exist a morphism $f:X \to \mathbb{P}^1$ and two infinite subsets $A$ and $B$ of $\mathbb{P}^1(k)$ such that the preimages of $A$ and $B$ by $f$ lie inside $S$ and $T$ respectively?

Let $X$ be a smooth projective curve over an algebraically closed field $k$. Let $S$ and $T$ be two disjoint infinite subsets of $X(k)$. Does there exist a morphism $f:X \to \mathbb{P}^1$ and two infinite subsets $A$ and $B$ of $\mathbb{P}^1(k)$ such that the preimages of $A$ and $B$ by $f$ lie inside $S$ and $T$ respectively?

Edit: As pointed out by Dragon, this may fail when $k$ is countable, but I would like to know the answer in the general case.

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Jérôme Poineau
Jérôme Poineau
  • 4.1k
  • 1
  • 23
  • 35

Partitions of algebraic curves and maps to $\mathbb{P}^1$

Let $X$ be a smooth projective curve over an algebraically closed field $k$. Let $S$ and $T$ be two disjoint infinite subsets of $X(k)$. Does there exist a morphism $f:X \to \mathbb{P}^1$ and two infinite subsets $A$ and $B$ of $\mathbb{P}^1(k)$ such that the preimages of $A$ and $B$ by $f$ lie inside $S$ and $T$ respectively?