I am trying to understand $\mathbb Q$-Gorenstein smoothings, and especially 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 $\psi \colon \mathcal X \rightarrow \Delta$ 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 isomorphic to $X$,
- 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 a family $\psi \colon \mathcal X \to \Delta $ as above such that $X'=\psi^{-1}(t)$ for some $t\in \Delta$.
For example, I am trying to understand the following.
Question 1. Assume that there exists a fixed integer $m$ such $mK_{X_t}$ is Cartier for every $t$. Is there any example where 3rd condition is not satisfied?
Question 2. If the total space of the family is $\mathbb Q$-Gorenstein, is it true that the 3rd condition is satisfied?
References with examples would be appreciated.