TITLE: Geometric motivation for the Stanley-Reisner correspondence
QUESTION [10 upvotes]: The Stanley-Reisner ring of an abstract simplicial complex $\Delta$ on the vertex set $\{1,...,n\}$ is the $k$-algebra
$$
k[X_1,...,X_n]/I_\Delta
$$
where $I_\Delta$ is the ideal generated by the $X_{i_1}...X_{i_r}$ with ${i_1,...,i_r}\notin \Delta$.
Somebody told me that this construction helps to study varieties $k[X_1,...,X_n]/I$ for an arbitrary ideal $I$ as follows (if I am not missing something): Let $X_1<...< X_n$ be a monomial order and consider the initial ideal $I':=in_{<}(I)$ of $I$. The passage from $k[X_1,...,X_n]/I$ to $k[X_1,...,X_n]/I'$ is called 'flat deformation' and this term makes sense if I draw pictures of the varieties. Many properties of $I$ (like dimension) are directly related to properties of $I'$.
The aim is to find a to $I'$ related ideal $I_\Delta$ for an abstract simplicial complex $\Delta$.
I was told that the problem that $I'$ has a generator like $X_1^2$ could be resolved by introducing a new variable $X_1'$, replacing $X_1X_1$ by $X_1X_1'$ and mod out $X_1-X_1'$ of $k[X_1,...,X_n,X_1']$. First, I don't understand why this should be closer to the form $I_\Delta$ because $I_\Delta$ is generated by monomials.
My main question is: 

Can one see in a concrete affine
  example how the geometry of $\Delta$
  relates to the initial variety $I$?

If you take for example the simplicial complex $\Delta$ (I apologize that I can not typeset the brackets) $\emptyset,X, Y, Z, XY, XZ, YZ$ which looks like a one sphere, the associated variety is the union of the $XY$, the $XZ$ and the $YZ$ hyperplane in $\mathbb{R}[X,Y,Z]$. Why is this reasonable? On the other hand, I would like to know how the simplicial complex (after the transformation indicated above) corresponding to the circle $I=(X^2+Y^2-1)$ in $\mathbb{R}[X,Y,Z]$ looks like. It would be nice if this is associated to the $\Delta$.
Please tell me, if this question is completely unreasonable or doesn't make any sense. I am an absolute beginner in algebraic geometry.

REPLY [7 votes]: I think you're asking if there's a direct geometric relationship between an algebraic variety $X=V(I)$ (i.e., the zero set of an ideal $I$) and the Stanley-Reisner complex $\Delta_I$ or $\Delta_{in(I)}$ --- in other words, does the variety look like the simplicial complex?  In general, I think the answer can be quite subtle, and is probably best approached using the algebra as an intermediary..  For example, if $\Delta$ is a simplicial sphere, then the number $d$ of its facets will correspond to the degree of $X$, but you will not necessarily be able to ``see'' directly from $X$ how those facets fit together combinatorially in $\Delta$ (e.g., if $d=20$, is $\Delta$ a decagonal bipyramid or an icosahedron?).  On the other hand, you can calculate $d$ and similar invariants fairly easily from the ideal $I$ (in this case, compute the Hilbert series as a rational function in $q$, and plug $q=1$ into the numerator).
For "flat deformation", here's the example I always keep in mind (very similar to Torsten's): take the hyperbola $xy=c$ and let $c\to0$.  In the limit, the hyperbola degenerates into a pair of lines $x=0$ and $y=0$.  This corresponds to replacing the ideal $(xy-c)$ (or $(xy-cz^2)$, if you want to think projectively) with the monomial ideal $(xy)$, which is its initial ideal under the right term ordering on $\Bbbk[x,y,z]$.  The general principle is that some invariants (e.g., Hilbert series) don't change when you degenerate, and some (e.g., singularities) can only get \emph{worse}, so if you can prove that the degeneration is, say, Cohen-Macaulay then so was the original thing.
As the operation of replacing, e.g., $X_i^2\in I$ by $X_iX_i'$, it is called ``polarization'' and has similarly mild effects.  See, e.g., \S3.2 of Miller and Sturmfels \emph{Combinatorial Commutative Algebra}.