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This question is from Rick Miranda's book:

$C$ is the curve in $\mathbb{P}^3$ defined by $f=x_0x_3-x_2x_1=0$ and $g=x_0^2+x_1^2+x_2^2+x_3^2=0$

The question asks to prove that it is a complete intersection and also asks to found the topologic genus. My doubt is:

This point $P=[x_0=1,x_1=i,x_2=i,x_3=-1]$ satisfies both equations, and the jacobian in this point:

$$\left(\begin{array}{rrrr}-1&-i&-i&1\\2&2i&2i&-2\end{array}\right)$$ shows that $rank\neq 2$ in that point. So, does this means that the curve is not a complete intersection? Anyway, even with this points where the curve is singular, does it has genus 1?

This has already reviewed on several previous posts, but I still have this doubt, I will appreciate if anyone could help me solve it.

Protágoras
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    https://math.stackexchange.com/questions/21164/problem-in-rick-miranda-finding-genus-of-a-projective-curve?rq=1 – 54321user Jun 11 '17 at 17:16
  • So, I understand that it must be a typing error on the book, the equation should be: $f=x_0x_3-2x_1x_2$. Thanks! – Protágoras Jun 11 '17 at 18:35

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