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Let $X$ be a smooth quasi-projective variety over a field $k$, with pure dimension $d$, $K/k$ an arbitrary field extension, and $\xi\in Z^1(X)$ an algebraic cycle on $X$ of codimension $1$.

  • DoesFor any algebraic cycle $\eta$ of codimension $1$ on $X_K$ ($\eta\in Z^1(X_K)$), does there exist an algebraic cycle $\eta\in Z^1(X_K)$$\xi\in Z^1(X)$ (ie. defined over $k$) such that $\xi_K-\eta$ is effective?

  • Fix a collection $\{Y_1,\ldots, Y_n\}$ of $k$-subvarieties of $X$, and further assume each component of $\xi$$\eta$ intersects each of the $Y_i$$(Y_i)_K$'s in prescribed fixed codimension $p$. Does there exist $\eta\in Z^1(X_K)$$\xi\in Z^1(X)$ such that each component of $\eta$$\xi$ intersects each of the $(Y_i)_K$$Y_i$'s in codimension $p$, and $\xi_K-\eta$ is effective?

Let $X$ be a smooth quasi-projective variety over a field $k$, with pure dimension $d$, $K/k$ an arbitrary field extension, and $\xi\in Z^1(X)$ an algebraic cycle on $X$ of codimension $1$.

  • Does there exist an algebraic cycle $\eta\in Z^1(X_K)$ such that $\xi_K-\eta$ is effective?

  • Fix a collection $\{Y_1,\ldots, Y_n\}$ of $k$-subvarieties of $X$, and further assume each component of $\xi$ intersects each of the $Y_i$'s in prescribed fixed codimension $p$. Does there exist $\eta\in Z^1(X_K)$ such that each component of $\eta$ intersects each of the $(Y_i)_K$'s in codimension $p$, and $\xi_K-\eta$ is effective?

Let $X$ be a smooth quasi-projective variety over a field $k$, with pure dimension $d$, $K/k$ an arbitrary field extension.

  • For any algebraic cycle $\eta$ of codimension $1$ on $X_K$ ($\eta\in Z^1(X_K)$), does there exist an algebraic cycle $\xi\in Z^1(X)$ (ie. defined over $k$) such that $\xi_K-\eta$ is effective?

  • Fix a collection $\{Y_1,\ldots, Y_n\}$ of $k$-subvarieties of $X$, and further assume each component of $\eta$ intersects each of the $(Y_i)_K$'s in prescribed fixed codimension $p$. Does there exist $\xi\in Z^1(X)$ such that each component of $\xi$ intersects each of the $Y_i$'s in codimension $p$, and $\xi_K-\eta$ is effective?

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user92332
user92332

Effective cycles of codimension 1 and field extensions

Let $X$ be a smooth quasi-projective variety over a field $k$, with pure dimension $d$, $K/k$ an arbitrary field extension, and $\xi\in Z^1(X)$ an algebraic cycle on $X$ of codimension $1$.

  • Does there exist an algebraic cycle $\eta\in Z^1(X_K)$ such that $\xi_K-\eta$ is effective?

  • Fix a collection $\{Y_1,\ldots, Y_n\}$ of $k$-subvarieties of $X$, and further assume each component of $\xi$ intersects each of the $Y_i$'s in prescribed fixed codimension $p$. Does there exist $\eta\in Z^1(X_K)$ such that each component of $\eta$ intersects each of the $(Y_i)_K$'s in codimension $p$, and $\xi_K-\eta$ is effective?