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.

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.