I will start the story by the end:

Is there some caracterization of (some of) the singularities arasing from the Minimal Model Program (canonical, terminal, log-...) in terms of commutative algebra ?

The general motivation is that everyone says that they are "mild" singularities, and I would like to know if, for instance, can we deduce that a variety with canonical singularities which is also Gorenstein (and $\mathbb{Q}$-factorial?) is a local complete intersection or not?

The original motivation is that I was looking at some prime nef divisor $D$ in a terminal, $\mathbb{Q}$-factorial, Gorenstein, Fano variety of dimension $n\geq 3$, and I wanted to prove that $D$ it wasn't ample. Finally I arrived to prove that the restriction map
$$\operatorname{H}^2(X,\mathbb{R}) \to \operatorname{H}^2(D,\mathbb{R}) $$
it wasn't an injection. But now the problem is that we know that the Lefschetz hyperplane theorem is true if, for instance, $X\backslash D$ is smooth or $X$ is a local complete intersection (see [this MO question](http://mathoverflow.net/questions/57744/lefschetz-hyper-plane-theorem-for-singular-projective-varieties), for example), which is not my case.

More generally, it has been proven by Goresky and MacPherson in their book "Stratified Morse theory" that if we define a measure $s(p)$ of the degree of singularity of $X$ at a point $p$ to be (the number of equations needed to define $X$ near $p$) minus (the codimension of $X$ in projective space), the we have an analogue

> **Theorem** (the LHT for singular spaces): Let $X$ be a purely $n$-dimensional algebraic subvariety of complex projective space, and let $H$ be a generic hyperplane. Then $H_i(X,X\cap H)=0$ for $i<n-\sup_{p\in X} s(p)$.

So again it arises the question that if we could analyse how far is a variety with "mild" singularities to be a local complete intersection, namely, compute $s(X)=\sup_{p\in X} s(p)$ ?

Thank you very much in advance for your comments and references.