Let $X$ be a complex surface and $X^{[n]}$ be the Hilbert scheme of finite analytic subspaces $Z$ for which $dimH^0(Z,\mathcal{O}_Z)=n$. I have trouble understanding $X^{[n]}$. That's what i've worked out:

If we take $n$ distinct points $p_1,p_2,\cdots,p_n$ there is nothing to see ($I_Z=\{f\in\mathcal{O}_X|f(p_1)=\cdots f(p_n)=0\}$)

If two points coincide ($p_1=p_2$) then i think the subspace $Z$ will have a defining ideal of the form $I_Z=\{f\in \mathcal{O}_Z|f(p_1)=0,df_{p_1}v=0,f(p_3)=0,\cdots,f(p_n)=0\}$ for a certain $v\in T_{p_1}X$. Then if $X^{(n)}$ is the symmetric n-product of $X$ and $h:X^{[n]}\rightarrow X^{(n)}$ is the Hilbert-Chow morphism it is that $h^{-1}((p_1,p_1,p_3,\cdots,p_n))\simeq \mathbb{P}^1$ i.e. varying the direction of $v\in T_{p_1}(X)$ we obtain different points in $h^{-1}((p_1,p_1,p_3,\cdots,p_n))$.

But now if the three points coincide how is $I_Z$ defined (i can not understand it)? what is the fiber of $h$ above $(p_1,p_1,p_1,p_4,\cdots,p_n)$? I suppose it's "bigger" than $\mathbb{P}^1$.. How is the case of $i$ coinciding points ($3<i\le n$)?

Thank you very much, i hope this is not a too trivial question, i'm just approaching the study of the Hilbert scheme of points, so i will be open to suggestions for references.