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?



Effective cycles of codimension 1 and field extensions
The answer to the both questions is no. Consider $X=\mathbb{A}^1$, $k=\mathbb{Q}$, and $K=\mathbb{C}$ (any transcendental extension will do here). Let $\eta=\{\pi\}$. The cycles on $X\_K$ of the form $\xi\_K$ are precisely the finite Galois-stable linear combinations of elements of $\bar{\mathbb{Q}}\subset\mathbb{C}$, so $\xi\_K-\eta$ can never be effective.


