# Examples of non-trivial closed subschemes of a complete non-projective non-singular variety

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.

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I don't understand this question. All closed subschemes come from some ideal sheaf? –  Alex Youcis Mar 20 '14 at 7:13
@AlexYoucis In other words, I'm asking an example of a non-trivial ideal sheaf whose support is a given closed subset. –  Makoto Kato Mar 20 '14 at 9:01
But, what makes it non-trivial? If I am understanding your post, you are discounting all ideal sheaves coming from closed subschemes? –  Alex Youcis Mar 20 '14 at 9:03
@AlexYoucis I'm asking an example of ideal sheaf other than $\mathcal I^n$. –  Makoto Kato Mar 20 '14 at 9:06
But it sounds like you're talking about $\mathcal{I}^n$ for any $\mathcal{I}$ coming from a closed subscheme, which is all possible $\mathcal{I}$! –  Alex Youcis Mar 20 '14 at 9:07

Take the subscheme consisting of a double point and a simple point !

Edit
As requested in the comments, here are a few more details.
Choose two closed points $x,y\in V$.
Let $\mathcal I(x)\subsetneq \mathcal O_V$ and $\mathcal I(y)\subsetneq \mathcal O_V$ be the ideal sheaves of these two points considered as reduced closed points of $V$.
Then define the ideal sheaf $\mathcal K\subset \mathcal O_V$ to be the unique sheaf of ideals of $\mathcal O_V$ satisfying:
a) $\mathcal K|(V\setminus \{y\})=\mathcal I(x)|(V\setminus \{y\})$
b) $\mathcal K|(V\setminus \{x\})=\mathcal I(y)^2|(V\setminus \{x\})$
If $\mathcal I$ is the ideal sheaf of the reduced subscheme $Z=\{x,y\}$ we then have $$\cdots\subsetneq \mathcal I^n\subsetneq \cdots \mathcal I^2 \subsetneq \mathcal K \subsetneq \mathcal I \subsetneq \mathcal O_V$$ so that $\mathcal K$ defines a non-reduced subscheme $Z'$ whose reduction is $Z$ and satisfying $$Z=Z_1\subsetneq Z'\subsetneq Z_2\subsetneq \cdots\subsetneq Z_n\subsetneq\cdots V$$ so that $Z'$ is distinct from the all the thickenings $Z_n$ of $Z$ defined by the $\mathcal I^n$.

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Georges, how precisely does one rigorously define this? Clearly in local coordinates you mean $V(x_0^2,x_1)$, but I don't know how to define this for a general scheme. Thanks! –  Alex Youcis Mar 20 '14 at 10:09
Dear @Alex, define the ideal $\mathcal I|U$ on some affine open subscheme $U$ in which you choose two closed points and extend that ideal by $\mathcal O_X$ outside of the two closed points. –  Georges Elencwajg Mar 20 '14 at 10:15
[and extend that ideal by $\mathcal O_X$ outside the two closed points] I'm afraid I don't understand this. Could you explain how precisely you define this ideal sheaf on $X$? –  Makoto Kato Mar 20 '14 at 19:50
Thanks for the nice answer. By the way, your example makes me wonder if there is an example of a non-trivial subscheme of $V$ whose underlying topological space is irreducible. –  Makoto Kato Mar 21 '14 at 6:17
Dear Makoto, you are welcome. Yes, I think there are irreducible examples: you should take a reduced subscheme and fatten just one point on it. This would correspond in commutative algebra to an embedded prime ideal in the primary decomposition of an ideal. But I haven't done any calculation, so I can't formally guarantee that it works. –  Georges Elencwajg Mar 21 '14 at 7:11