Theorem (C-Sertöz)
The quartic surface $X : x^4 + xyzw + y^3 z + yw^3 + z^3 w = 0 \subset \mathbf{P}^3$ has $\operatorname{Pic} \overline{X} = \Lambda$, generated by quartics over a quadratic extension of $L := \mathbf{Q}(\{a_i\}_i)$.
Wanna be a Theorem (C-Sertöz)
There is a practical algorithm to compute the saturation of the lattice generated by rational curves of degree up to 4.
KEEP SLIDE: reused verbatim from an existing talk.
SOURCE: L:L851-870
CORRECTION candidate (unapproved, minima-41h / minima-rza): "The quartic surface $X : x^4 + xyzw + y^3 z + yw^3 + z^3 w = 0 \subset \mathbf{P}^3$ has $\operatorname{Pic} \overline{X} = \Lambda$, the saturation of the lattice spanned by the 336 Galois-conjugate smooth rational quartics, which are all defined over the normal closure of $K/\mathbf{Q}$."
PANEL FLAG: the index 343 is asserted from the author's computation and was not independently checkable by the panel.
Two words carrying the weight. The field: a quadratic extension of L defines the two components of a single quadric section, but the Galois conjugates of those curves are only simultaneously defined over the normal closure, which is the field the Galois action of slides 16 and 17 is computed over. The saturation: the 336 curves generate a full-rank sublattice of index 343 in Pic Xbar, so integral generation requires an explicit saturation step, which is precisely what the wanna-be theorem below asks for.
Spoken: A quadratic extension of $L = \mathbf{Q}(\lbrace a_i \rbrace)$ splits one quadric section; the whole conjugate family needs the normal closure.
Spoken: And the 336 curves span only a sublattice of index 343 , so the saturation is not cosmetic.
TRANSCRIBED FROM: five-nomial-quartics/slides/mukai_leiden.tex:851-870 (the frame's closing line 868 is deck slide 19)