Let $k$ be an algebraically closed field. A variety over $k$ is a separated integral scheme of finite type over $k$. Let $V$ be a complete non-projective non-singular variety over $k$. Let $Z$ be a closed subset of $V$. Let $\mathcal I$ be the ideal sheaf which defines $Z$ as a reduced closed subscheme of $V$. $\mathcal I^n$ defines a closed subscheme $Z_n$ of $V$ for every integer $n \ge 1$. I would like to know examples of closed subschemes of $V$ other than $Z_n$.
Remark The more examples, the better. Please don't think that the question would be solved if one example would be given.
Edit(March 23, 2014) I have just posted a similar question in MathOverflow.