Let $X$ be a (smooth complex algebraic) surface. Suppose $\theta$ is an automorphism of order $2$ of $X$, such that its fixed locus is a disjoint union of smooth curves. I am trying to prove that the quotient 
$$Y=X/\langle\theta\rangle$$
is in fact a smooth surface. (First of all: is this true/does it follow from some general result?)


My attempt:


Since the quotient is clearly smooth away from the fixed points, we can localize the question around a fixed point $p=(0,0)\in \Bbb{C}^2$. Up to a chart change we can suppose that the fixed curve is the $z$-axis and
$$\theta(z,w)=(z,-w)$$
$\Bbb{C}[z,w]^\theta$ has two generators $Z=z^2$ and $W=w$.
Now the smoothness of the quotient amounts to show that in fact there are no relations between $Z$ and $W$ and thus the quotient around $p$ is locally the spec of $\Bbb{C}[z\_1,z\_2]$. But I have no idea how to do this.


A secondary question is: is it always $\rho(Y)=\rho(X)$ (Picard number) ?



Smoothness of the quotient surface by an involution with nice fixed locus
Francesco gave the answer to your first question. As for the second one, look at my answer to [this post](https://mathoverflow.net/questions/149875/k3-surface-with-a-non-symplectic-involution-a-basic-question).


