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 $X=z^2$ and $Y=w$. Now the smoothness of the quotient amounts to show that in fact there are no relations between $X$ and $Y$ 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.