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.