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Added a positive answer under the assumption that $\dim S = 1$.
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Let $S$ be a smooth affine variety over $\mathbb C$ and let $f:X\to S$ be a finite unramified morphism.

Suppose that $X(K(S))$ is non-empty. (This means that $X\to S$ has a section generically. It does not imply $X\to S$ being generically trivial.)

Is $X(S)$ non-empty?

The answer is positive if $S$ is of dimension at most one.

Let $S$ be a smooth affine variety over $\mathbb C$ and let $f:X\to S$ be a finite unramified morphism.

Suppose that $X(K(S))$ is non-empty. (This means that $X\to S$ has a section generically. It does not imply $X\to S$ being generically trivial.)

Is $X(S)$ non-empty?

Let $S$ be a smooth affine variety over $\mathbb C$ and let $f:X\to S$ be a finite unramified morphism.

Suppose that $X(K(S))$ is non-empty. (This means that $X\to S$ has a section generically. It does not imply $X\to S$ being generically trivial.)

Is $X(S)$ non-empty?

The answer is positive if $S$ is of dimension at most one.

Source Link
Unit section
Unit section
  • 21
  • 3

Are generically trivial finite unramified morphisms trivial

Let $S$ be a smooth affine variety over $\mathbb C$ and let $f:X\to S$ be a finite unramified morphism.

Suppose that $X(K(S))$ is non-empty. (This means that $X\to S$ has a section generically. It does not imply $X\to S$ being generically trivial.)

Is $X(S)$ non-empty?