I want to show that up to automorphisms, the number of parameters of the intersection of three general quadrics in $\mathbb{P}^4$ is exactly $12$. I want to do it using the tools of chapter $7$ by Miranda and no schemes.
My idea is: we have 24 parameters describing automorphisms of $\mathbb{P}^4$ and $\binom{6}{2}=15$. So $24-15=9$ and we can fix one of these quadrics. How can I complete the proof?