I am trying to understand definition of $\mathbb Q$-Gorenstein smoothing, specially the third condition in the following definition.
Definition: For a normal projective surface $X$ with quotient singularities, a $\mathbb Q$-Gorenstein smoothing is a one-parameter flat family of projective surfaces $\mathcal X \rightarrow \Delta$ over a small disk $\Delta$, which satisfies the following three conditions:
- the general fibre $X_t$ is a smooth projective surface,
- the central fibre $X_0$ is $X$,
- the canonical divisor $K_{\mathcal X / \Delta}$ is $\mathbb Q$-Cartier.
We say that $X'$ is a $\mathbb Q$-Gorenstein smoothing of $X$ if there exists such an $\mathcal X$ and $X'=\Psi^{-1}(t)$ for some $t\in \Delta$.
For example, I am trying to understand the following.
Q1: Let $m$ be fixed integer such $mK_{X_t}$ is Cartier for every $t$. Is there any example where 3rd condition is not satisfied.
Q2. If the total space of the family is $\mathbb Q$-Gorenstein then is it true that 3rd condition is satisfied.
Moreover, is there any reference with examples to understand this.