Let $X$ be a smooth projective geometrically connected variety over a field $k$.
Let the cyclic group $G=\{e,a\}$ with two elements act on $X \times X$ via $a\cdot (x_1,x_2) = (x_2,x_1)$.
When is the quotient $X\times X/ G$ nonsingular?
I wrote down the case of $X=\mathbf{A}^1$. The answer is that $X\times X/G$ is given by nonsingular scheme $\mathrm{Spec} (k[xy,x+y]) \cong \mathbf{A}^2$, where $\mathbf{A}^1_x\times \mathbf{A}^1_y = X\times X = \mathrm{Spec} (k[x]\otimes k[y])$. (The subscript $x$ and $y$ indicate the coordinate used.)
I'm very interested in the case $\dim X = 1$.