Reconstructing quadric surfaces

$$X : x^{4} + xyzw + y^{3}z + yw^{3} + z^{3}w = 0 \subset \mathbf{P}^3$$

$$\operatorname{Pic}(\overline{X})|_B \subseteq \Lambda \overset{?}{\subseteq} \operatorname{Pic}\overline{X}$$

• Fortunately, there is a small $\operatorname{Aut}(\Lambda)$ orbit of size 336:
  $133056 = 336 + 1008 + 1176 + 3528 \cdot 3 + 4704 \cdot 3 + 7056 \cdot 9 + 14112 \cdot 3$
• For each quartic curve $C \subset X$, we can compute

  $$I_{[C],2} = \langle a_0 x^2 + \cdots + a_9 w^2 \rangle_{\mathbf{C}}$$

  that defines a quadric surface $Q$, such that $Q \cap X = C \cup \overline{C}$. Hence, we expect an orbit of 168 quadrics each containing a pair of quartics.
• We aim reconstruct the ten (algebraic!) coefficients of these quadrics.