Consider three quadratics in $\mathbb{C}P^4$:
$$
x_0^2+x_1^2+x^2_2=0,~ x_1x_4+x_2x_3=0, ~ x_0^2+x_3^2+x_4^2=0.
$$

If there intersection *was* non-singular, then the intersection should be a curve of genus $5$, see [this note][1]. 

However, the intersection in our problem has $4$ singular points
$$
[0,1,\pm i,0,0] \text{ and } [0,0,0, 1, \pm i].
$$

So it seems to me that (the normalization of) the intersection curve should have genus
$$
g=5-4=1.
$$

I want to know if my guess is correct and if it is part of some general result.


  [1]: https://chenhi.github.io/math6670-f19/solutions/hw10-4.pdf