Suppose that $S$ is a non-singular complex projective surface that is defined over $\overline{\mathbb Q}$, namely $S\cong\text{Proj}\frac{\mathbb C[T_1,T_2,\ldots,T_n]}{(f_1,\ldots,f_n)}$ where $f_i$ has coefficients in $\overline{\mathbb Q}$ for every $i$. If $S'$ is a minimal model of $S$, then can we conclude that also $S'$ is defined over $\overline{\mathbb Q}$? Does the blow-down preserve the field of definition (for algebraically closed fields)?

Suppose that $S$ is a non-singular complex projective surface that is defined over $\overline{\mathbb Q}$, namely $S\cong\text{Proj}\frac{\mathbb C[T_1,T_2,\ldots,T_n]}{(f_1,\ldots,f_n)}$ where $f_i$ has coefficients in $\overline{\mathbb Q}$ for every $i=1,\ldots,n$. If $S'$ is a minimal model of $S$, then can we conclude that also $S'$ is defined over $\overline{\mathbb Q}$? Does the blow-down preserve the field of definition (for algebraically closed fields)?

Suppose that $S$ is a non-singular complex projective surface that is defined over $\overline{\mathbb Q}$, namely $S\cong\text{Proj}\frac{\mathbb C[T_1,T_2,\ldots,T_n]}{(f_1,\ldots,f_n)}$ where $f_i$ has coefficients in $\mathbb Q$. If $S'$ is a minimal model of $S$, then can we conclude that also $S'$ is defined over $\overline{\mathbb Q}$? Does the blow-down preserve the field of definition (for algebraically closed fields)?

Suppose that $S$ is a non-singular complex projective surface that is defined over $\overline{\mathbb Q}$, namely $S\cong\text{Proj}\frac{\mathbb C[T_1,T_2,\ldots,T_n]}{(f_1,\ldots,f_n)}$ where $f_i$ has coefficients in $\overline{\mathbb Q}$ for every $i$. If $S'$ is a minimal model of $S$, then can we conclude that also $S'$ is defined over $\overline{\mathbb Q}$? Does the blow-down preserve the field of definition (for algebraically closed fields)?