Let $X$ be a smooth projective variety and $\gamma\in CH\_i(X)$ be an effective cycle. Suppose that $\gamma$ is supported on $Y$ in the sense that $\gamma\in \ker(CH\_i(X)\to CH\_i(X\setminus Y))$. Then by the localization exact sequence $\gamma$ is in the image of $CH\_i(Y)\to CH\_i(X)$. There is a priori no reason for $\gamma$ to be the image of an effective cycle. What is an example where $\gamma$ does not come from an effective cycle? 



Does an effective cycle supported on a subvariety come from an effective cycle on that subvariety
Let $C$ and $D$ be (integral) curves in $\mathbb{P}^2$ of degrees $2$ and $3$ respectively. Consider the inclusion


$$i\colon C \cup D\hookrightarrow\mathbb{P}^2.$$


The image in $CH\_1(\mathbb{P}^2)$ of the set of effective cycles in $CH\_1(C\cup D)$ is


$$\mathbb{Z}\_{\geq0} \ 2H + \mathbb{Z}\_{\geq0}\ 3H$$


and hence does not contain the class $H$ of a line. However, $H = i\_\*([D]-[C])$ is supported on $C\cup D$ and effective.


