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.



Some examples of $\mathbb Q$-Gorenstein smoothing
**Question 1**. The answer is *yes*, and the classical example is as follows. Is it possible to find a one-parameter family $\psi \colon \mathcal{X} \to \Delta$ such that $X\_0$ is isomorphic to the cone over a rational normal curve $C\_4 \subset \mathbb{P}^4$, whereas $X\_t (t \neq 0)$ is isomorphic to a smooth rational normal scroll $X \subset \mathbb{P}^5$. 


The surface $X\_0$ is $\mathbb{Q}$-factorial, because it has only a cyclic quotient singularity of type $1/4(1, \, 1)$. However, your third condition is not satisfied since for all $t \neq 0$ we have $$9=K\_{X\_0}^2 \neq K\_{X\_t}^2=8,$$ whereas (as an immediate consequence of Condition 3) any $\mathbb{Q}$-Gorenstein smoothing preserves the self-intersection of the canonical class.


**Question 2**. The answer is still *yes*. If $\mathcal{X}$ is $\mathbb{Q}$-Gorenstein, then there exists an integer multiple $mK\_{\mathcal{X}}$ of it which is a Cartier divisor. But therefore the divisor $$mK\_{\mathcal{X}/ \Delta}= mK\_{\mathcal{X}} -m \psi^\* K\_{\Delta}$$
is also Cartier.


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**References.** 


* The deformations of cones over rational normal curves are described for instance in


J. Stevens, *Deformations of Singularities*, Lecture Notes in Math. **1811**, Springer 2003.


* For the theory of $\mathbb{Q}$-Gorenstein smoothing of surface singularities, see


M. Manetti, *Smoothing of singularities and deformation types of surfaces*, in *Symplectic 4-manifolds and algebraic surfaces*, 169–230,
Lecture Notes in Math. **1938**, Springer 2008


and the references given therein.


