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edit approved Apr 16, 2019 at 20:47
user137767
edit approved Apr 16, 2019 at 20:47
user137767

Let $X$ be the affine line with a double origin over Spec $\mathbb Z$$\mathrm{Spec}\,\mathbb Z$. Let $X_\eta$ be its generic fibre, the affine line with a double origin over Spec $\mathbb Q$$\mathrm{Spec}\,\mathbb Q$.

Let $0$ be one of the origins of $X_\eta$.

Are there infinitely many ways of extending $0$ to a section of $X$?

My feeling is that besides taking the closure of $0$ in $X$, we can also alter our choice of origin at some primes. If that feeling is correct, then we would get infinitely many sections of $X\to $ Spec $\mathbb Z$$X\to \mathrm{Spec}\,\mathbb Z$ inducing the same section generically.

Let $X$ be the affine line with a double origin over Spec $\mathbb Z$. Let $X_\eta$ be its generic fibre, the affine line with a double origin over Spec $\mathbb Q$.

Let $0$ be one of the origins of $X_\eta$.

Are there infinitely many ways of extending $0$ to a section of $X$?

My feeling is that besides taking the closure of $0$ in $X$, we can also alter our choice of origin at some primes. If that feeling is correct, then we would get infinitely many sections of $X\to $ Spec $\mathbb Z$ inducing the same section generically.

Let $X$ be the affine line with a double origin over $\mathrm{Spec}\,\mathbb Z$. Let $X_\eta$ be its generic fibre, the affine line with a double origin over $\mathrm{Spec}\,\mathbb Q$.

Let $0$ be one of the origins of $X_\eta$.

Are there infinitely many ways of extending $0$ to a section of $X$?

My feeling is that besides taking the closure of $0$ in $X$, we can also alter our choice of origin at some primes. If that feeling is correct, then we would get infinitely many sections of $X\to \mathrm{Spec}\,\mathbb Z$ inducing the same section generically.

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Shane
Shane
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In how many ways can one extend the zero section of the affine line with a double origin

Let $X$ be the affine line with a double origin over Spec $\mathbb Z$. Let $X_\eta$ be its generic fibre, the affine line with a double origin over Spec $\mathbb Q$.

Let $0$ be one of the origins of $X_\eta$.

Are there infinitely many ways of extending $0$ to a section of $X$?

My feeling is that besides taking the closure of $0$ in $X$, we can also alter our choice of origin at some primes. If that feeling is correct, then we would get infinitely many sections of $X\to $ Spec $\mathbb Z$ inducing the same section generically.