I've come across a double cover of $\mathbb P_1(\mathbb C)$, ramified at $[1:1]$ and $[-1:1]$ in homogeneous coordinates, given as the quotient by the natural $\mathbb Z/2\mathbb Z$-action generated by $[x:y]\mapsto[y:x]$. This space is totally natural and must have a name! For instance, it can be obtained as the space of all ratios $\lambda_1/\lambda_2$ of eigenvalues of non-nilpotent $2\times2$ complex matrices (which is the case I'm interested in).

In the litterature the notion of "reduced symmetric product" sometimes arises as $M\times M\setminus\Delta$, where $\Delta$ is the diagonal. This is somehow related, but in the case I'm confronted to I want to keep the branch-point (corresponding to $\Delta\setminus\{(0,0)\}$ in the case $M:=\mathbb C$, before projectivization). Also I feel the context is not completely the same.

My question: how should I call/write $\mathbb P_1(\mathbb C)~/~\mathbb Z/2\mathbb Z$? Is there a standard denomination? Notation?

**Edit**: as Noam D. Elkies very rightly mentions in a comment, the quotient is conformally equivalent to $\mathbb P_1(\mathbb C)$, *via* the explicit parameterization $[x^2+y^2:xy]$. It doesn't settle the question of whether this particular perception of the projective line has a particular name, though. But maybe this is not interesting.

Loïc Teyssier's double cover has two branch points (as expected by Riemann-Hurwitz): the involution $(x:y) \leftrightarrow (y:x)$ fixes not just $(1:1)$ but also $(-1:1)$. $\endgroup$ – Noam D. Elkies May 26 '15 at 3:26