For $n = 3$ the variety $x_1 + x_2 + .. + x_n = 0$, $x_1 x_2 .. x_n = 1$ is an elliptic curve, and for $n = 4$ it is rational [edit: or so I thought, before seeing the other replies].

What can be said, for example regarding rationality, for larger $n$ (for values of $x_i$ where the variety has no components of the same form with smaller $n$)? Is this variety of a standard type, such as toric or Calabi-Yau?

One might assume that it remains rational for $n > 4$; but, given that the degree of the product increases, that might well not be so.