Smoothing of Stanley-Reisner type singularities (Q1177789)

Let \(P_ \ell\) the set of all subsets of \(\{1,\ldots\ell\}\) and \(A\subset P_ \ell\). We say that \(A\) is closed if \(\{\emptyset\}\in A\) and if \(\alpha\in A\), \(\beta\in A\) and \(\alpha\subset x\subset\beta\), then \(x\in A\). It exists a bijection between the closed part of \(P_ \ell\) and the reduced subschemes of \(\mathbb{C}^ \ell\) invariant under the standard action of the torus \(T=(\mathbb{C}^*)^ \ell\). This bijection associates to \(A\), \(X(A)=\text{Spec}\mathbb{C}[x_ 1,\ldots,x_ \ell]/(I)\) where \(I=(x_{i_ 1}\ldots x_{i_ k}\mid\{i_ 1\ldots i_ k\}\notin A)\). --- We say that \(X(A)\) is spherical if \(A\) is spherical as an abstract simplicial complex. The paper gives some necessary or sufficient, but distinct, conditions for the smoothability of \(X(A)\). For example, let \(A\) be associated to the triangulation \(T\) of the homological sphere of dimension \(n-1\), then 1. if \(n=2\), \(X(A)\) is smoothable 2. if \(n=3\) and \(\text{Ext}^ 2_{{\mathcal O}_ x}(L^ x,{\mathcal O}_ x)=0\) \((L^ x\) is the cotangent complex), then \(X=X(A)\) is smoothable if and only if \(T\) is not an icosahedron.