Picard lattice of a K3 surface

Goal

From the equations of $X$, compute $\operatorname{Pic} \overline{X} \subset H_2(X, \mathbf{Z})$ as a $\operatorname{Gal}(\bar k/k)$-module.

Recall Lecture 1: we are doing this via a running example.

SOURCE: L:L472-496, compressed

The number-field hypothesis is doing work. Hochschild-Serre gives the isomorphism only once the next obstruction H^3(k, G_m) vanishes, which holds over a number field; over a general base one has an exact sequence rather than an isomorphism.

What it does. Names what "computing the Picard lattice" has meant all along, and makes clear that Lectures 1 and 2 were only ever computing one number of it.

SECTION 3.3, Picard lattices of K3 surfaces. Fourteen slides, the destination of the course. Candidate classes, then curves, then a certified lattice, ending on what is still open.

Spoken: The target, stated once and properly: compute $\operatorname{Pic} \overline{X}$ as a $\operatorname{Gal}(\bar k/k)$-module, not merely its rank.

TRANSCRIBED FROM: five-nomial-quartics/slides/mukai_leiden.tex:472-496