$$\begin{aligned} &Q : a_0 x^2 + a_1 x y + \cdots + a_9 w^2 = 0 \subset \mathbf{P}^3, \quad [L := \mathbf{Q}(\{a_i\}_i):\mathbf{Q}] = 168\\ &\Lambda_Q := \langle [C] : C \subset \sigma(Q) \cap X, \, \sigma : L \hookrightarrow \mathbf{C} \rangle \subseteq \operatorname{Pic}(\overline{X})|_B \subseteq \Lambda \overset{?}{\subseteq} \operatorname{Pic} \overline{X} \end{aligned}$$
The inclusion $\Lambda_Q \subseteq \Lambda$ is not explicit!
Nonetheless, $\operatorname{Pic} \overline{X}$ and $\Lambda$ are saturated in $H_2(X, \mathbf{Z})$.
Hence, it is sufficient to show that $\operatorname{rank} \Lambda_Q = \operatorname{rank} \Lambda = 19$.
KEEP SLIDE: reused verbatim from an existing talk.
SOURCE: L:L755-791
What it does. The destination of the whole course. Note what it delivers that Lectures 1 and 2 could not: not a number, but the lattice.
Say why the upper bound is there. Saturation plus a common rank-19 sublattice does not exclude Picard rank 20, with Lambda a corank-one saturated sublattice of Pic Xbar. The bound rank Pic Xbar <= 19 closes that off: Lambda_Q sits inside both Lambda and Pic Xbar with rank 19, so the two saturated lattices have the same rational span and coincide.
PANEL FLAG: the bijection and the count of 336 curves do not by themselves give rank 19. A bijection of two sets of size 168 is compatible with the 336 classes spanning any rank up to 19, so the rank computation is a separate step, and the supplied sources do not show it being performed. State it as a requirement, not as done.
The two counts are at different scales, and the source's one-line version runs them together. 66528 = 133056/2 bounds the quadrics globally, one per residual pair of quartic classes. The bijection is the local statement: 168 embeddings sigma of L into C, 168 conjugate quadrics sigma(Q), 168 pairs of quartic classes, 336 curves. It is not a bijection between 66528 quadrics and 168 pairs.
HEDGE, as written in the source: "Verbatim, including: 'The inclusion Lambda_Q is contained in Lambda is not explicit!' Then the rescue: both Pic Xbar and Lambda are saturated in H_2(X,Z), so it suffices that rank Lambda_Q = rank Lambda = 19."
The frame closes the display Pic Xbar = Lambda with a certificate stamp image and a check mark; the deck carries no images, so the stand-in line names the file.
Spoken: With $\operatorname{rank}\operatorname{Pic}\overline{X} \leq 19$ from slide 9, all three lattices have rank 19, so $\Lambda$ and $\operatorname{Pic}\overline{X}$ are saturated with the same rational span, hence equal.
Spoken: Two routes to the rank equality: intersect the 336 curves with each other over $\mathbf{F}_p$; or, over $\mathbf{C}$, bound the quadrics arising from the 133056 classes by $66528 = 133056/2$, one per residual pair, and match the 168 conjugates $\sigma(Q)$ bijectively with the 168 pairs of quartic classes they cut out.
Spoken: Either route still owes the rank itself: an explicit certificate that $\operatorname{rank} \Lambda_Q = 19$, a nonzero $19 \times 19$ minor of the intersection matrix of the 336 classes, or a prior verification that those 336 vectors span $\Lambda$.
Spoken: With that, $\operatorname{Pic} \overline{X} = \Lambda$, with the certificate mark.
TRANSCRIBED FROM: five-nomial-quartics/slides/mukai_leiden.tex:755-791