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:

1. the general fibre $X_t$ is a smooth projective surface,
2. the central fibre $X_0$ is $X$,
3. 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.

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:

1. the general fibre $X_t$ is a smooth projective surface,
2. the central fibre $X_0$ is isomorphic to $X$,
3. 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.

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:

1. the general fibre $X_t$ is a smooth projective surface,
2. the central fibre $X_0$ is $X$,
3. 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.

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:

1. the general fibre $X_t$ is a smooth projective surface,
2. the central fibre $X_0$ is $X$,
3. 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.