@user6818: To reiterate earlier replies, the Riemann surface constructed from slit planes is smooth and compact. Your authors' equation should be viewed as defining a locus in $S^2 \times S^2$, viewed as the componentwise compactification of $\mathbb{C}^2$. In this setting:

1. As Alexandre Eremenko and Paul Garrett noted, the "Riemann surface of $\sqrt{z}$" is the sphere $S^2$, equipped with the squaring mapping $f(w) = w^2$, namely, the (compact, smooth) locus of $z = w^2$ in $S^2 \times S^2$. The branch points (images of critical points of $f$) are $z = 0$ and $\infty$. Since $f$ is two-to-one, the classical construction is to take two copies of $\mathbb{C}$ (with coordinate $z$), make a slit between the branch points (e.g., by removing the negative real axis), and to "cross-glue" the edges of the slits. Here, the result is another copy of $\mathbb{C}$ (with coordinate $w$). Adding the branch point $z = \infty$ compactifies the $w$ plane to $S^2$.

2. Igor Khavine's example of the "(complex) circle" $w^2 + z^2 = 1$ (again, a compact, smooth locus in $S^2 \times S^2$) may be viewed as "the Riemann surface of $\sqrt{1 - z^2}$". The picture is identical to the squaring map (rotate the "$z$ sphere" to bring $0$ and $\infty$ to $-1$ and $-1$, say); the branch points are $\pm 1$, and the Riemann surface can be constructed by slitting two copies of $\mathbb{C}$ along the closed interval $[-1, 1]$ and "cross-gluing" the edges. In the diagram below, the slits are the upper and lower halves of the circle in the vertical plane above the real axis; the two "sheets" ($z$ planes) of the surface are the two "tilted planes". The locus is smooth even at the branch points $z = \pm 1$, i.e., $w = 0$. (The vertical axis is the real part of $w$. At risk of posting gratuitous eye candy, there's an animation loop at http://mathcs.holycross.edu/~ahwang/epix/misc/complex/quadric.gif that shows the result of rotating the $w$ plane while keeping $z$ fixed.)

Andy

(I don't have enough reputation to post images, but the diagram is at https://i.sstatic.net/kfyKN.png. I also can't post comments, so instead am answering. If you're a helpful passerby and can fix these issues, please feel free to edit accordingly.)

@user6818: To reiterate earlier replies, the Riemann surface constructed from slit planes is smooth and compact. Your authors' equation should be viewed as defining a locus in $S^2 \times S^2$, viewed as the componentwise compactification of $\mathbb{C}^2$. In this setting:

1. As Alexandre Eremenko and Paul Garrett noted, the "Riemann surface of $\sqrt{z}$" is the sphere $S^2$, equipped with the squaring mapping $f(w) = w^2$, namely, the (compact, smooth) locus of $z = w^2$ in $S^2 \times S^2$. The branch points (images of critical points of $f$) are $z = 0$ and $\infty$. Since $f$ is two-to-one, the classical construction is to take two copies of $\mathbb{C}$ (with coordinate $z$), make a slit between the branch points (e.g., by removing the negative real axis), and to "cross-glue" the edges of the slits. Here, the result is another copy of $\mathbb{C}$ (with coordinate $w$). Adding the branch point $z = \infty$ compactifies the $w$ plane to $S^2$.

2. Igor Khavine's example of the "(complex) circle" $w^2 + z^2 = 1$ (again, a compact, smooth locus in $S^2 \times S^2$) may be viewed as "the Riemann surface of $\sqrt{1 - z^2}$". The picture is identical to the squaring map (rotate the "$z$ sphere" to bring $0$ and $\infty$ to $-1$ and $-1$, say); the branch points are $\pm 1$, and the Riemann surface can be constructed by slitting two copies of $\mathbb{C}$ along the closed interval $[-1, 1]$ and "cross-gluing" the edges. In the diagram below, the slits are the upper and lower halves of the circle in the vertical plane above the real axis; the two "sheets" ($z$ planes) of the surface are the two "tilted planes". The locus is smooth even at the branch points $z = \pm 1$, i.e., $w = 0$. (The vertical axis is the real part of $w$. At risk of posting gratuitous eye candy, there's an animation loop at http://mathcs.holycross.edu/~ahwang/epix/misc/complex/quadric.gif that shows the result of rotating the $w$ plane while keeping $z$ fixed.)

Andy