Let $F_1, F_2$ and $F_3$ be finites sets of irreducible polynomials in $\mathbb{C}[x_0, \ldots, x_n]$ of degree at most $2$ such that $F_1 \cap F_2 \cap F_3 = \varnothing$ and for every $Q_1, Q_2$ from distinct sets there is a polynomial $P$ in the remaining set such that $V(Q_1, Q_2) \subseteq V(Q)$.
Is it true that $\text{trdeg}_{\mathbb{C}}(\cup_i F_i) = O(1)?$
This is a special case of Conjecture 1 here. Such questions are important for algebraic complexity.
It is more or less clear how to solve such question for polynomial of degree $1$ (see, for example, https://arxiv.org/abs/1211.0330) however I have not intuition for quadratic polynomials.
Could you give some tools, ideas that can prove/disprove it?
