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 $\mathbb{Q}$-Gorenstrein 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.

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.

${}$

${}$ 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.

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.

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.

${}$

${}$ 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 $\mathbb{Q}$-Gorenstrein 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.