3.3 Picard lattices of K3 surfaces
Twelve spoken slides in the proposed version, the destination of the course. Candidate classes, then curves, then a certified lattice, ending on the Galois questions and a request for the next surface. Current slide 20 is written out below, but marked as the proposed drop.
8 Picard lattice of a K3 surface
L:L472–496, compressed
The target, stated once and properly: compute Pic X̄ as a Gal(k̄/k)-module, not merely its rank. With one line on why the Galois structure matters: for k a number field, H¹(Gal(k̄/k), Pic X̄) ≃ Br₁(X)/Br₀(X).
>>[minima-uv3.230] AUTHOR (2026-09-15): Slide 8 is overlaps Lecture 1, so we should make it clear. and might also be in Lecture 2, so we do not need to repeat everything. But just say that we are doing via a running example
[C] The overlap is total, and it is the same source frame twice. Every one of the four things on this slide is already printed in Lecture 1:
[C] - Setup and curve-quotient display: Lecture 1 deck position 2, "The geometric Picard group", has $X/k$ a K3 surface; $k\subset\Bbb{C}$ a number field and Pic(X^al) \simeq \Z\langle algebraic curves in X^al \rangle / \langle linear equivalences \rangle \subset H_2(X_C, \Z). [C] - T(X) splitting: Lecture 1 deck position 4, "Pic inside $H^2$", has H^2(X_C,\Q) \simeq Pic(X^al)_\Q \oplus T(X)_\Q and, as its own bullet, The "new and interesting" Galois representations arise from $T(X)$. Same sentence, same quotation marks as here. [C] - Goal box, Zariski quotation and Brauer-Manin: Lecture 1 deck position 5, "Computing Pic as a Galois module", carries all three. Its Goal reads From the equations of $X$, compute Pic(X^al) \subset H_2(X_C,\Z)$ as a $Gal(k^al/k)$-module, word for word this slide's, and both Brauer displays sit in a Corollary box. [C] - Provenance: artifacts/plan/lecture1.md gives slide 4 the locator "I15:L151-173; L:L472-496", and this entry's locator is "L:L472-496, compressed". mukai_leiden.tex has three consecutive "Picard lattice of a K3 surface" frames, at 444, 472 and 499; Lecture 1 took 444-449 and 472-496, and this slide takes 472-496 over again. The repetition is not two slides converging on the same facts, it is one frame transcribed twice.
[C] Lecture 2 checked, and it is clear. "Brauer", "Br_1", "Zariski" and "grave difficulties" have zero hits in both artifacts/plan/lecture2.md and talks/picard-lattices-2.html. Lecture 2's Picard material is the finite-field kernel form on its slide 8, Pic(X_Fp)_Qp = ker(Frob_p - p), and the Hodge filtration on its slide 5; neither is on this slide. So "we do not need to repeat everything" bites against Lecture 1 alone. Lecture 2 does duplicate Lecture 1, but on two different slides, "Picard lattice, over finite fields" and "Reduction to finite characteristic", which artifacts/plan/plan.md line 123 already records.
[C] Lecture 1 already promises this slide out loud, which makes the recall easier than it looks. The last spoken note on Lecture 1 deck position 5 is "The Galois module matters for rational points through the Brauer-Manin obstruction. In Lecture 3 we compute it for an actual surface." So the shape you are asking for is half built already: Lecture 1 states the goal and promises the example, and this slide is where the promise is kept.
[C] Concrete form. Title unchanged, three lines, everything else off: a recall naming Lecture 1 and carrying only the Goal and H^1(Gal(k^al/k), Pic X^al) \simeq Br_1(X)/Br_0(X) as things already seen; then the turn, that this lecture actually does it rather than stating it; then the handoff to the running example, which is slide 9's quartic x^4 + xyzw + y^3z + yw^3 + z^3w = 0. The definitions, the lattice display, the Zariski quotation and the T(X) splitting all come off, since a room that sat through Lecture 1 this morning has seen each of them.
[C] Consequence, and it collides with two things already open on this lecture. The first item under "Open on this lecture" asks whether the Brauer-Manin material stays visibly here as in the deck, or is deferred as the old written page says; this comment settles that toward removing the visible displays, because they are Lecture 1's. But the entry's own prose says "The Brauer-Manin payoff is deferred to slide 20, where H1(Gal, Pic X-bar) comes back as an open question", and slide 20 is the PROPOSED DROP, still unapproved and tracked as minima-uv3.2. If slide 8 sheds the Brauer material and slide 20 is dropped, then Br_1/Br_0 is stated on Lecture 1 slide 5, promised there, and never returned to anywhere in the course. Recommend that if 20 goes, the one-line payoff either stays here or moves to slide 17, which already ends on "what is H1(Gal, Pic X-bar)".
[C] What is this slide's own and worth keeping through the cut. Two things here are not in Lecture 1. The number-field hypothesis on the Brauer isomorphism, that Hochschild-Serre gives it only once H^3(k, G_m) vanishes and that over a general base one gets an exact sequence rather than an isomorphism, appears in this plan's prose and is not visible on any Lecture 1 slide; it is a footnote at most. The second is the entry's own "What it does." line, that Lectures 1 and 2 were only ever computing one number of the lattice. Once the definitions go, that sentence is arguably what the slide is for.
The number-field hypothesis is what the isomorphism needs, not decoration: Hochschild-Serre gives it only once the next obstruction H³(k, G_m) vanishes, which holds over a number field. Over a general base one gets an exact sequence instead of an isomorphism.
The Brauer-Manin payoff is deferred to slide 20, where H¹(Gal, Pic X̄) comes back as an open question; here it would be a name with nothing attached to it.
Deck (governs). The visible slide does not defer the payoff: it defines Pic X̄ inside H₂, states the Galois-module Goal, gives the Picard/transcendental splitting, and displays both the Brauer-group isomorphism and the Brauer-Manin inclusions.
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.
9 A running example inspired by Klein-Mukai
L:L556–607
X : x⁴ + xyzw + y³z + yw³ + z³w = 0 ⊂ P³
A fibre in a pencil of generic rank 19, so rank Pic X̄ ≥ 19; the methods of Lectures 1 and 2 give the matching upper bound rank Pic X̄ ≤ 19, which is needed later and is worth stating now. Heuristically Λ ≃ Z¹⁹ with ΠΛ ≈ 0. Then Aut Λ ≃ F₄₂ × PGL(2,7), no lines, no conics, no twisted cubics, and lattice computations predict 133056 smooth rational quartics spanning Λ.
>>[minima-uv3.231] AUTHOR (2026-09-15): We should break Slide 9 in two slides. We should also add teh lattice for Pic X. It is in some paper of Voight about https://jvoight.github.io/articles/CommonFactor-imj-fixed-errata-031825.pdf but the point is that when we put this in a toric variety the ambient picard lattice embeds n the picard of the family. Maybe astra can fiugre this out, is doable in sage.
[C] The paper, identified and read. CommonFactor-imj-fixed-errata-031825.pdf is Charles F. Doran, Tyler L. Kelly, Adriana Salerno, Steven Sperber, John Voight and Ursula Whitcher, "Zeta functions of alternate mirror Calabi-Yau families", in the errata-fixed version dated 18 March 2025, 29 pages.
[C] Your surface is in it, exactly. Table (5.1.1) lists five invertible K3 pencils in P^3 with the same dual weights, and the fourth is F1L3 = x_0^4 + x_1^3 x_2 + x_2^3 x_3 + x_3^3 x_1 - 4 psi x_0 x_1 x_2 x_3, with symplectic symmetry group Z/7Z. Substituting x_0 = x, x_1 = y, x_2 = z, x_3 = w gives x^4 + y^3 z + z^3 w + w^3 y - 4 psi xyzw, which is this slide's x^4 + xyzw + y^3z + yw^3 + z^3w at psi = -1/4. So the "pencil" in the first bullet is F1L3 and this X is its fibre at psi = -1/4. That is worth putting on the slide: the pencil is currently named only as "a pencil", with no equation and no citation.
[C] What the paper does give you: the rank, with a proof. Section 5.2: "For the five pencils in Table 5.1.1 with dual weights (1, 1, 1, 1), we conclude that the generic Neron-Severi rank is at least 22 - 3 = 19; but it cannot be 20, because then the family would be isotrivial, so it is equal to 19." That is a citable source for this slide's first bullet, which today asserts generic rank 19 with nothing attached.
[C] What the paper does not give you: the lattice. Immediately after, in its own words: "The complete Neron-Severi lattice of rank 19 for the case of the Dwork pencil F4 is worked out via transcendental techniques by Bini-Garbagnati [BG14, section 4]. It would be interesting to compute the full Neron-Severi lattices for the remaining four plus two families; Kloosterman [Kl17] has made some recent progress on this question." F1L3 is one of the remaining four. As of March 2025 these authors describe computing its lattice as open, not as done. Full reference for the one that is done: G. Bini and A. Garbagnati, "Quotients of the Dwork pencil", J. Geom. Phys. 75 (2014), 173-198.
[C] The toric statement is also not in this paper. "Toric" occurs four times in it, all incidental: Candelas-de la Ossa-Rodriguez-Villegas in the introduction, a change to toric coordinates inside the Dwork-cohomology setup twice, and one bibliography entry. There is no statement that the ambient Picard lattice embeds in the Picard lattice of the family. That construction is real and standard, but it comes from the toric hypersurface literature (Batyrev-Cox for the cohomology of hypersurfaces in toric varieties, and the lattice-polarized toric K3 line of work for the Picard lattice itself), not from here.
[C] So the comment splits into a citation and a computation, and only the first is a lookup. The citation, that the pencil is F1L3 and its generic rank is 19, is available now from the paper above. The lattice is a computation nobody in this reference chain has published for F1L3.
[C] Feasibility in Sage, since you asked. The route is real: build the toric ambient from the Newton polytope of the five monomials, take its Picard lattice with the intersection form induced by the anticanonical class, and restrict to the hypersurface. Sage has the pieces natively (lattice polytopes with PALP, ToricVariety, and divisor class groups with intersection numbers), so this is a script rather than a research program. Two things make it not merely mechanical: the ambient lattice gives the toric part of the Picard lattice, and whether it is all of it needs the non-toric classes ruled out; and the embedding gives a finite-index sublattice, so the saturation is a separate step.
[C] That second caveat is the one to keep in view, because it is the same gap as slide 15's. The generic member's lattice specializes into Pic of this fibre with finite index, and equal rank does not give equal lattice. This lecture already carries an asserted index for exactly this kind of step: slide 18's "the 336 curves generate only a sublattice of index 343 in Pic X-bar". Noting without claiming anything: 343 = 7^3, and the symplectic symmetry group of F1L3 in Table 5.1.1 is Z/7Z. Whether those are the same 7 is worth an hour of somebody's time, not a sentence on a slide.
[C] Payoff, if the computation lands. An explicit Gram matrix for Lambda would discharge or unblock five beads that are currently all waiting on you for the same missing object: minima-rza (Aut Lambda = F_42 x PGL(2,7) and the 133056 count, both asserted from your computation and uncheckable by the panel) and minima-uv3.64 (no degree-four square-zero class, which is what excludes the (2,2)+(2,2) splitting on slide 14) would be settled outright; minima-uv3.66 (the separation bound covering norm -6 on slide 3), minima-uv3.65 (the index 343) and minima-uv3.63 (the rank-19 certificate on slide 15) all need it as a precondition. That is the strongest single reason to run it.
[C] Direct consequence for the slide as written. The deck's fourth bullet is "No known explicit descriptions of Pic X-bar". Adding the lattice makes that bullet false, so it comes off in the same edit rather than sitting next to its own refutation.
[C] Proposed split, which falls out of the content rather than being imposed. The nine bullets divide cleanly at the point where the surface stops being described and the lattice starts predicting:
[C] - 9a, the surface and its rank: the equation; that it is the fibre of F1L3 at psi = -1/4, with the citation; generic rank 19 hence rank Pic X-bar >= 19; the matching upper bound <= 19 from Lectures 1 and 2; and then the new lattice, replacing "No known explicit descriptions". [C] - 9b, what the lattice predicts: the heuristic Lambda ≃ Z^19 with Pi.Lambda ≈ 0 and the inclusion chain; Aut Lambda ≃ F_42 x PGL(2,7); no lines, conics or twisted cubics; smooth rational quartics as the smallest curves; and the 133056 prediction as the closing line.
[C] The 133056 should end 9b rather than sit mid-list, since the plan's own "What it does." says that number is the hook for the rest of the lecture. Splitting here is what lets it land as a punchline instead of the ninth bullet on a crowded slide.
[C] Count consequence. This lecture's stated target is 19 spoken slides, and the only route to it on the page is dropping slide 20, which is still unapproved (minima-uv3.2). Splitting 9 puts the proposed count at 20, or 21 once the title slide of minima-uv3.227 is added. Same arithmetic as Lecture 2's slide-2 expansion: the split is affordable but has to be paid for somewhere, and the first item under "Open on this lecture" no longer reaches 19 on its own.
[C] On dispatching astra: the brief is ready to go and I have not sent it, since this pass is recording rather than acting. It would need the equation, psi = -1/4, the target (an explicit Gram matrix for the rank-19 lattice, with the toric part identified and the saturation stated separately), the five beads above as the acceptance criteria, and the instruction to return the matrix rather than prose. Say go and it leaves.
>>[minima-uv3.233] AUTHOR (2026-09-15): We shuld add a further details about F1L3 on the lattice, I can always remove them later
[C] Scope. Follow-on to minima-uv3.231 above, same slide, and it lands on 9a under the split proposed there. Taking "I can always remove them later" literally, what follows is a generous menu with a locator on every item rather than a recommended selection. Everything below is from Doran-Kelly-Salerno-Sperber-Voight-Whitcher, "Zeta functions of alternate mirror Calabi-Yau families", 18 March 2025, section 3 of which is devoted to this family precisely because, in their words, "it is of particular interest, and has rather special features".
[C] Where the 19 comes from, which is the item that most earns its place. Section 5.2 derives it in one move: the five pencils of Table 5.1.1 share a degree three Picard-Fuchs equation for the holomorphic form, equation (5.1.2), which after a change of variables is the hypergeometric equation for 3F2(1/4, 1/2, 3/4; 1, 1; psi^-4). Three is therefore the rank of the transcendental part, so the generic Neron-Severi rank is 22 - 3 = 19, "but it cannot be 20, because then the family would be isotrivial, so it is equal to 19". The slide currently asserts generic rank 19 flat. For a room of algebraic geometers, "the Picard-Fuchs equation has order 3, so the transcendental lattice has rank 3" is one line and it converts an assertion into a reason.
[C] The symmetry, from Lemma 3.1.2 with n = 3. Set m = n^n + (-1)^(n+1) = 28. The group of diagonal scalings preserving F_psi is cyclic of order 28; the subgroup acting trivially on X_psi is the cyclic group of order n + 1 = 4; so the group acting faithfully and symplectically is cyclic of order 28/4 = 7. That is the Z/7Z in the third column of Table 5.1.1.
[C] And that 7 is the same 7 the slide is already full of, which is the connection worth making explicitly. The deck says Aut Lambda ≃ F_42 x PGL(2,7), and this plan's own note unpacks F_42 = AGL(1,7), of order 42 = 7 . 6, with |PGL(2,7)| = 336 and 42 . 336 = 14112. Klein's quartic curve, which the family is named for, has orientation-preserving automorphism group the simple group of order 168 = |PSL(2,7)|, index 2 in PGL(2,7). Slide 18 asserts the 336 curves generate a sublattice of index 343 = 7^3. So the symplectic Z/7Z of the pencil, the F_42, the PGL(2,7) and the 343 are plausibly one phenomenon seen four times, and saying so out loud is cheaper than any of the four numbers individually.
[C] Provenance of the name, which is a free sentence and a good one. From the introduction: "The pencil takes its name from Klein's quartic curve, whose group of orientation-preserving automorphisms is isomorphic to the simple group of order 168, and the member of the family F1L3 at psi = 0, which appears as an extremal example during Mukai's classification of finite groups of automorphisms of K3 surfaces that preserve a holomorphic form". The slide's title already says "inspired by Klein-Mukai" and never says why; that quotation is the why, and it names both halves.
[C] Smoothness and bad primes, from Lemma 3.1.3: for p not dividing m = 28 and psi^(n+1) = psi^4 != 1, the fibre is smooth, nondegenerate and convenient with respect to x_n. So the singular fibres are exactly the fourth roots of unity and the bad primes are 2 and 7. Checked against this slide: your surface is the fibre at psi = -1/4, and (-1/4)^4 = 1/256, so it is a smooth member. Worth having in the notes if anyone in the room asks why 7 is excluded from the reductions.
[C] Mirror symmetry, if you want a bonus line. Let H_psi be the Greene-Plesser mirror family of quartics in P^3, the fibrewise quotient of the Dwork pencil F4 by (Z/4Z)^2 resolved. A computation of Kadir, quoted at (1.3.2), gives Z(H_psi, T) = 1 / ((1 - T)(1 - qT)^19 (1 - q^2 T) R_psi(T)), with the 19 appearing directly as an exponent. Corollary 1.3.3 then says that for p != 2, 5, 7 the five pencils, F1L3 among them, have the same zeta function as H_psi after a finite extension of F_q. So the 19 on your slide is also visible on the mirror side, which is the kind of remark this audience will enjoy and which costs one sentence.
[C] Recommendation on how much to take, since you asked for more rather than less. The Picard-Fuchs order 3 giving 19 is the one I would not cut: it is the only item that turns an asserted number on the slide into an argument. The Klein and Mukai quotation is the second, because the title already promises it. The Z/7Z is the third, because it connects to Aut Lambda two bullets later. The smoothness lemma belongs in the speaker notes and the mirror-symmetry line is genuinely optional.
[C] One consequence of adding sourced material here. Everything above carries a citation; the two numbers next to it, Aut Lambda ≃ F_42 x PGL(2,7) and the 133056 prediction, still do not, and are the subject of minima-rza's PANEL FLAG. Putting cited facts beside uncited ones makes the uncited ones more conspicuous rather than less, which is an argument for running the computation in minima-uv3.231 rather than against adding this detail.
>>[minima-uv3.235] AUTHOR (2026-09-15): go, dispatch astra for the lattice, I do not remember all the details of the paper. that is why it would be nice to write it on the slide. and we should ask gpt-6 astra to review it
[C] Dispatched and running, on the second attempt. Bead minima-uv3.234, "L3 s09: compute the Picard lattice of the F1L3 fibre via the toric ambient". Wrapper codex-usyd, model gpt-6-astra, model_reasoning_effort=xhigh, service_tier=fast, per fleet-pick top --force :gpt-6-astra. Brief at artifacts/orch/lattice-astra-brief.md, report to artifacts/orch/lattice-astra.md. The brief carries the equation, psi = -1/4, the toric route, and the five beads above as acceptance criteria, and requires the toric part, the question of whether it is all of Pic, and the saturation index to be reported separately rather than conflated. The first attempt went to codex-usyd, which fleet-pick top --force :gpt-6-astra selected, and exited 1 with "ERROR: Your workspace is out of credits. Ask your workspace owner to refill in order to continue." before doing any work. The author then said, verbatim: "We are now using codex! please fix that in the instructions that still point you to codex-usyd". ~/.agents/fleet-routing.toml was listing every codex-usyd row ahead of its personal-codex twin; the four pairs and the four lane lists are now reordered so personal codex leads, with codex-usyd kept as the fallback and a dated comment saying why. fleet-pick now returns row top-astra on wrapper codex, and the run is live.
[C] Your reason changes what this material is for, which changes where it goes. "I do not remember all the details of the paper. that is why it would be nice to write it on the slide" makes the F1L3 detail a speaker aid, not decoration for the room. Minimal faithful reading applied: visible on the slide, as you said, with whatever does not fit going to the speaker notes rather than being cut.
[C] It also supersedes my own recommendation in minima-uv3.233. I proposed trimming that menu to three items and putting the smoothness lemma in the notes. Under the reason you have now given, the trim is wrong: the value is in having the details in front of you, so the fuller list is the operative instruction and my "what I would not cut" line above should be read as ranking, not as a cut list.
[C] "Ask gpt-6 astra to review it" is ambiguous between two objects, and one reading breaks a recorded rule. If "it" is the lattice computation, that is astra reviewing its own work, which artifacts/orch/RULES.md forbids: "a THIRD agent reviews THAT COMMIT", and "A writer checking its own work verifies intent rather than what was asked". If "it" is the F1L3 detail written onto the slide, that is a different object from astra's computation and there is no conflict. Taking the second reading, since it is the one that follows your sentence order, "that is why it would be nice to write it on the slide, and we should ask astra to review it", and since it does not require breaking a rule. So: a second astra pass checks the slide text against the paper, dispatched once that text exists. If you meant the computation, that review goes to sol instead, per the cross-family rule, and I will send it on one word.
[C] What astra's review would check, when the text exists. Every F1L3 claim on the slide against its locator in the paper: the order-three Picard-Fuchs equation and 22 - 3 = 19 with the isotriviality step, section 5.2; the symplectic Z/7Z from Lemma 3.1.2 with m = 28 and the order-4 scalars; the Klein and Mukai naming sentence from the introduction; smoothness at psi^4 != 1 and bad primes 2 and 7 from Lemma 3.1.3; and the mirror line (1 - qT)^19 at (1.3.2), attributed to Kadir. That is a verification pass against a fixed document, which is the kind of review a single model can do well.
>>[minima-uv3.236] AUTHOR (2026-09-15): We should add somewhere that is example started at some point about a weekend workshop at ICERM about thinking about K3 surfaces on the LMFDB (this should be asy to find somewhere)
[C] Found, with the exact details. The workshop is "K3 Surfaces in the LMFDB", 14-17 July 2024, at ICERM, Brown University, Providence, Rhode Island, sponsored by the Simons Foundation and ICERM. Source: the LMFDB's own activities page, https://www.lmfdb.org/acknowledgment/activities, which lists it verbatim in that form. Reported elsewhere as a Simons Collaboration focused activity organised by Noam Elkies, Adam Logan and John Voight, with the stated purpose of bringing families of K3 surfaces and their elliptic fibrations into the LMFDB; that organiser list is from secondary reporting rather than from the activities page, so treat it as the less certain half.
[C] Corroborated inside your own repository, which is presumably the "somewhere" you had in mind. five-nomial-quartics/paper/fivenomials.tex:106, in the Acknowledge section, reads "John Voight and the K3 meeting 2024 at ICERM", with Noam Elkies listed two lines above it. So the paper already credits the meeting; the deck does not mention it anywhere.
[C] One correction, because the room will contain people who were there. 14-17 July 2024 is a Sunday to Wednesday, four days, not a weekend. If the line is spoken as "a weekend workshop" it is checkably off. "A focused workshop at ICERM in July 2024 on getting K3 surfaces into the LMFDB" is both true and just as short.
[C] The line is better than it looks, because of where you are giving it. This minicourse is itself at ICERM, inside the semester programme "Computations on K3 Surfaces and Related Varieties", and the title slide's venue line already reads "ICERM: Arithmetic, Geometry and Computations on K3 surfaces". So the honest form of the remark is that the example started at ICERM in 2024 and is being presented at ICERM two years later, which lands with this particular audience in a way it would not anywhere else.
[C] Where to put it. Recommend slide 9a under the split in minima-uv3.231, as one line attached to the equation, since 9a is where the surface is introduced and its provenance is exactly what that slide is for. The alternative, if you would rather keep 9a purely mathematical, is the title slide as an acknowledgement, which is also where the Simons support line already sits. I would not put it on slide 18, the summary: arriving at the end it reads as a credit rather than as the story of where the example came from.
[C] Related, if you want the credit right rather than just the anecdote. The paper's acknowledgements are still a rough list, carrying "Stevell Muller (check spelling)", "Wim Nijgh (observations about 5nomials?)" and "Thomas Bouchet (for classifying the 5nomials)". Nothing on the deck needs that list, but if the workshop line names anyone, the names should match the ones that end up there.
>>[minima-uv3.244] AUTHOR (2026-09-15): Also, the talk in Sydney had some timings, we should perhaps add some of htose.
[C] Not in the archive. I searched every five-nomial source for runtimes, five-nomial-quartics/slides/mukai_sydney.tex and its duplicate under tex/, the other five mukai decks, paper/fivenomials.tex and emre_journal/. The only hit for any duration anywhere is the K2 mountaineering aside, "It got its first winter ascent 10 days ago", which appears in all of them. There is no computation timing in the Sydney deck as archived.
[C] One timing does exist in the wider source set, but it belongs to Lecture 2, not here: frobenious-dist/nyc-jnts.tex:1431, "Computing the zeta function or the Frobenius approximation for a prime takes several hours, thus trying to use 2 primes." That is the p-adic machinery, not the period or reconstruction work.
[C] Most likely the archived Sydney deck is not the version you are remembering, which fits what you said next about the paper having a newer version. If you can point at the current file, or just give me the numbers, they slot straight in. Where they would go: slide 12 already says "Every computation must be done extremely selectively!" and slide 14 says "Hopeless to do this directly! Operations in $L$ are seriously expensive!", both of which currently assert cost with no number attached. A timing on either turns an adjective into evidence, which is the register this audience responds to.
>>[minima-uv3.252] AUTHOR (2026-09-15): We should just say on the slides, that this also triggered new algorithm in mamga. maybe have symbol for it, to show, that the challenges we found have always been encountered from others on ways to improve, because before, no one thought we would try such computation
[C] The sentence already exists, in the current paper, and it is better than anything I would draft. five-nomial-current/paper/fivenomials.tex:68-69, verbatim: "Every software package we used struggled in many different ways upon contact with these computations. One challenge was the complexity of the base fields of the curves: these number fields had degrees in the hundreds with defining polynomials having coefficients of size reaching hundred thousands. We owe a great deal to the colleagues who stepped in to improve their algorithms as the work progressed; in our experience the computational environment is now materially more robust than it was when this paper began." That is your framing in your own words; the slide version is a compression of it, not a new claim.
[C] Inventory of what the badge would mark, so it is a real convention rather than one decoration. Five places, all documented in the source threads now in _sources/:
[C] - Slide 7. edgarcosta/endomorphisms, putatively in Magma by the end of the semester (minima-uv3.229). Your code going in, rather than someone else's being fixed, but the same story. [C] - Slide 12. Steel's new modular algorithm for matrices over number fields: M6's echelon form from 488 s to about 90 s, and his commitment that "Rank should always be at least as fast (and usually much faster) than EchelonForm! I will fix that in the Number Field case." [C] - Slide 12 again, and this is the strongest instance. A faster algorithm for factoring polynomials over a finite field when deflation is present, which Steel says is "really good in general", provoked by your degree-168 polynomial satisfying p(x) = g(x^7). [C] - Slide 14. Steel's own words about this deck's line: "Working hard to try to remedy your line 'Hopeless to do this directly! Operations in L are seriously expensive!'." [C] - Slide 17. The Elsenhans-Steel deflation paper in preparation, from "some polynomials that allow deflation" you supplied, plus Elsenhans's Galois-group work and his method for turning found groups into proven ones.
[C] There is glyph precedent in your own source, so this does not need inventing. mukai_leiden.tex sets images/certificatew.png inline four times at \includegraphics[height=1.5em], raised, with a checkmark alongside, as the certificate mark on the endomorphism theorem. A second inline mark at the same size would read as part of the same vocabulary rather than as an intrusion. Both certificate.png and certificatew.png are now in the extracted tree, which partly satisfies minima-uv3.67; k2_waqas_anees.jpg is still absent.
[C] On the symbol itself, a caution from earlier in this pass. If it is an emoji it inherits the problem I flagged under minima-uv3.239: the deck loads only Source Sans Pro, so an emoji falls through to whatever the projector machine has, and a tofu box in the middle of a slide is worse than no mark at all. A small PNG or inline SVG, sized like certificatew.png, is the safe form and matches the existing device. If you do want an emoji, a wrench or a gear reads as tooling improved without competing with the fearful faces on 13 and 14, which are doing the opposite job.
[C] The claim in your last clause is the interesting part and it is yours alone. "Before, no one thought we would try such computation" is not in the paper and no source I have states it; Steel's "I am also curious as where p comes from" and his surprise at the deflation factor are evidence for it, but they are not the claim. It reads well spoken and would need hedging if printed.
[C] Consequence worth weighing. A recurring badge turns five scattered anecdotes into a thread through the lecture, which is a real gain, but it also invites the room to count them. Four of the five are Magma, and three of those are Allan Steel, so the honest gloss is narrower than "software improved": your computations were the first inputs of this shape, and the people who own the tools responded. That is a better sentence anyway, since it is specific and checkable.
>>[minima-uv3.253] AUTHOR (2026-09-15): we should just say that orally
[C] Applied to the whole of the previous comment, not only its last clause. Nothing about the Magma algorithms goes on a visible slide; it becomes an oral line, and the symbol is dropped. If you meant only the unciteable "no one thought we would try such computation" and still want a printed line about the new algorithms, say so and it is one sentence.
[C] This has a recorded home rather than being a vague instruction. minima-uv3.177 carries your convention verbatim: "Let's also be sure to remove speaker notes, I really dont use those, except for anything I said in this conversations that I said I would say orally." So for Lecture 1 the notes were stripped to exactly the oral keep-list, ten passages across eight blocks. The Magma material is a Lecture 3 keep-list item on that same footing, and the precedent for the pattern is on that bead too: "we don't ave space on that slide at all, we will just say it out loud that is a discriminnat."
[C] Consequence for the five-place inventory above. It stays as the oral script, not as five badges. The two lines worth actually saying are Steel's own, since they are self-evidencing: that the echelon form went from 488 seconds to about 90, and that he is "working hard to try to remedy your line 'Hopeless to do this directly!'". Everything else in that inventory is context you would only reach for if the room asks.
>>[minima-uv3.254] AUTHOR (2026-09-15): We also developed a lot of sage code to handle this
[C] Evidenced, and stronger than "we wrote some scripts": there is an upstream contribution. five-nomial-current/ore_algebra-PR.md is a pull request against mkauers/ore_algebra fixing a real regression, that commit ba8b58e moved pitvdm_stdpts onto _fmpq_poly_interpolate_fmpq_vec, which needs FLINT >= 3.4, so on the FLINT 3.3.x still bundled with SageMath 10.8 the first import of ore_algebra.analytic.dac_sum_c dies with an undefined symbol. Your own words in it: "This in turn breaks any downstream package that touches dac_sum_c, e.g. lefschetz_family." It carries a deterministic reproducer and two candidate fixes, with a flint-pre-3.4-compat branch implementing the first.
[C] The local Sage side, from the archive listing. code/sagefix/ holds ore_algebra_ext, pylib, and packaged lefschetz_pylib at 0.1.19, 0.1.21 and a working copy; code/PeriodSuite/ is the period machinery with 86,957 generated ODE files under ode_storage; code/asphalt/ is 1,115 files of run directories; code/scripts/lefschetz/RUNBOOK.md documents the pipeline. So the claim on a slide would be that the method needed a patched and extended Sage stack, not merely a driver script.
[C] It also makes the previous comment symmetric, which is the better story. Magma improved because you handed Steel and Elsenhans inputs of a shape nobody had tried; Sage needed patching because the period machinery broke on contact with the same computation. Both directions, same cause. Said orally that is two sentences and it lands harder than the Magma half alone.
[C] One honest limit, since you also said the pipeline is human-intensive: your own words to Steel were that it "requires too many interactions between magma and sage, and to generate many examples I will need to streamline it". So the Sage code handled it rather than automated it, and a slide should not imply a finished toolchain.
>>[minima-uv3.246] AUTHOR (2026-09-15): ok, then skip it, I remember the rank computation took several hours per degree
[C] Taken as the number, so the archive hunt is dropped. "Several hours per degree" is your recollection, not a source citation, and it should be spoken that way rather than printed as a measured figure.
[C] Where it earns its place. Two slides currently assert cost with nothing attached: slide 12, "Every computation must be done extremely selectively!", and slide 14, "Hopeless to do this directly! Operations in $L$ are seriously expensive!". Recommend slide 12, because "several hours per degree" is about the rank computation over the reconstructed field, which is what slide 12 has just set up with its degree-168 minimal polynomials, and because slide 14's expense claim is then a callback rather than a second unquantified assertion.
[C] It also scales the whole lecture for the room, which the deck never does. Degrees 1 through 4, several hours each, is the cost of one surface; the corpus on slide 19 is every smooth unit-coefficient five-nomial quartic. That is the sentence that makes "every computation must be done extremely selectively" land as a constraint rather than a complaint.
[C] Consistency note, not a correction. The one timing in the wider source set is frobenious-dist/nyc-jnts.tex:1431, "Computing the zeta function or the Frobenius approximation for a prime takes several hours, thus trying to use 2 primes", which is Lecture 2's machinery. Two different computations, both "several hours", in two lectures of the same course. If both get said aloud, distinguish them, or the room will hear one number twice. PANEL FLAG. The isomorphism Aut Λ ≃ F₄₂ × PGL(2,7) and the predicted count 133056 are asserted from the author's lattice computation. The panel could not certify either from the supplied data; nothing has been changed.
Both symbols in that product need naming. F₄₂ = AGL(1,7) is the Frobenius group of order 42, not Z/7; the orders are what fix it, since 42 · 336 = 14112 = #Aut Pic X̄, the number slide 17 compares against. And Aut Λ means O(Λ, H), the isometries fixing the hyperplane class: −1 is an isometry of Λ and is not in the product, because it does not fix H.
[C] ASTRA RESULT, 2026-09-15, minima-uv3.234, full report at artifacts/orch/lattice-astra.md. VERDICT: DONE. It reached the rank-19 Picard lattice through the order-7 symmetry rather than the toric route, and it CONFIRMS both numbers on this slide: |O(Pic(X), H)| = 14112 from PARI qfauto and Magma AutomorphismGroup independently, and exactly 133056 classes with C.H = 4, C^2 = -2, from PARI and reproduced in Magma. It also confirms the orbit decomposition 336 + 1008 + 1176 + 3528*3 + 4704*3 + 7056*9 + 14112*3 = 133056, the unique 336-element orbit, its index 343, and a nonzero 19-by-19 minor of 645657712.
[C] But the GROUP STRUCTURE on this slide is refuted, while its ORDER stands. GAP returns C2 x (PSL(3,2) : (C7 : C6)), with centre of order 2. The obstruction is one line and I have checked it: r(H) = H and r(v) = -v for v in the orthogonal complement of H is a central isometry fixing H, so the group has non-trivial centre, whereas AGL(1,7) and PGL(2,7) both have trivial centre and so does their direct product. Orders agree, 42 * 336 = 14112 = 2 * 168 * 42, which is why the coincidence held up; the groups do not. Note this r is not the -1 that this plan's existing note already discusses, since -1 does not fix H and r does.
[C] Not yet verified, and it is the one step I would check before changing the slide: r is integral on the full Picard lattice only if it preserves the glue between ZH and its orthogonal complement. Astra asserts it does. If that holds the refutation is airtight. Also, astra computed the Picard lattice from the symmetry, not from your numerical candidate, which was not supplied to it; identifying its lattice with your Lambda is the missing link and astra says so explicitly rather than glossing it.
[C] Suggested slide wording if it holds: the deck currently hedges as Aut Lambda "seems to be" F_42 x PGL(2,7). The honest replacement keeps the number, which is the part that does work on the slide, and drops the product: order 14112, structure C2 x (PSL(2,7) : AGL(1,7)). Your call, and it needs your own check first. Default taken. The slide's \only<3-6> overlay recaps the even-rank fact, van Luijk, and the C-Sertöz p-adic step. With Lectures 1 and 2 behind us that overlay compresses to one back-reference line. This was flagged in the source map as the clearest cross-lecture edit, and it needed your approval; I have marked it rather than applied it.
Deck (governs). The back-reference compression is applied, Aut Λ is written as something that "seems to be" F₄₂ × PGL(2,7), and all eight facts, including the matching upper bound and 133056 prediction, remain visible.
What it does. The number 133056 is the hook. A prediction that large, from pure lattice theory, is what makes the rest of the lecture worth sitting through.
10 Reconstructing isolated curves from their Hodge classes
L:L609–628
Movasati-Sertöz
If γ = [C] for a curve C ⊂ X, then from (1/2πi)(∫_γ ω)_{ω ∈ F¹} one can construct an ideal I_γ with I(C) ⊂ I_γ.
Cifani-Pirola-Schlesinger
For a smooth rational quartic C ⊂ X, the quadric surface containing C generates I_{[C],2}.
Note F¹ reappearing here. This is the complex filtration, not Lecture 2's p-adic one; the two must not be run together, and saying so out loud costs a sentence.
>>[minima-uv3.232] AUTHOR (2026-09-15): We should break Slide 10 in two also, in particular it is a room full of algebraic geometry, so we might need to add some details, for example, no hope to recover rational curves of degree higher than 4. and we need "isolated" classes, so elliptic curves are also hard
[C] The details are already written, and not in the deck. five-nomial-quartics/paper/fivenomials.tex, 663 lines, is in the source archive and carries both points you name, in prose, with proofs. This plan declares one source prefix, "L, under artifacts/picard_minicourse/_sources/, is five-nomial-quartics/slides/mukai_leiden.tex", so the paper has no locator and nothing in Lecture 3 currently cites it. Adding a prefix for it is the cheapest way to make the rest of this comment executable. five-nomial-quartics/emre_journal/mukai.tex is a second unused file in the same directory.
[C] "Isolated", which is in your title and explained nowhere on the slide. The paper's argument, at paper/fivenomials.tex:312, verbatim: "The algebraic curve C is isolated in its cohomology class, i.e., there is no other effective algebraic 1-cycle D in [C] except D=C. Indeed, suppose C != D. But C and D are of the same degree and C is irreducible so C and D have no components in common. Hence C . D must be non-negative but D == C implies C . D = C . C = -2." It runs entirely on [C]^2 = -2, recorded just above at line 305 together with [C] . h_X = d. Three lines on the slide, and the title stops being a word the room has to take on trust.
[C] Elliptic curves, which is the same argument read backwards and is why it is worth stating. For a curve of arithmetic genus 1 adjunction gives E^2 = 2p_a - 2 = 0, so E . D = 0 is perfectly non-negative and the contradiction above evaporates. Concretely the class does not determine the curve: |E| is a pencil, the curve moves in a one-dimensional family, and there is no single C for the periods to reconstruct. So the method is not merely untested on elliptic curves, it has nothing to aim at. That is one sentence for an algebraic-geometry room and it makes "isolated" a hypothesis rather than an adjective.
[C] Degree at most 4, with the actual mechanism. paper/fivenomials.tex:414 states the Proposition: for C a smooth rational curve of degree d <= 4 and gamma = [C], I_{C,a} = I_{gamma,a} with a = 1 for d = 1,2 and a = 2 for d = 3,4. So the reconstruction only ever recovers the linear or the quadratic piece of the ideal, and the cutoff is where that piece stops determining the curve.
[C] FLAG: "no hope" is stronger than your own paper currently commits to, and the reason is commented out in the source. paper/fivenomials.tex:428 holds, inside a comment, the sentence that answers your question: "When d >= 5, we need to take a >= 3 to get a non-zero homogeneous component of I_{C,a}. In this case, the Jacobian ideal introduces superfluous equations, so the containment I_{C,a} subset I_{gamma,a} is strict." That is the mechanism. But it is commented out, and the co-author's margin notes next to it read "The situation for d >= 5 is commented out, it is interesting, check saturation of the Hodge ideal" and "But if we saturate the ideal generated by I_{gamma,a}, we should get rid of the Jacobian contribution since X is smooth." So the paper's own position is that saturating might remove the obstruction, which is why the paragraph was withheld. Applying your instruction as given, and saying so: the slide will assert on your authority something the paper deliberately did not. Recommend the hedged form, that above degree 4 the low-degree part of the Hodge ideal picks up superfluous Jacobian equations and the containment becomes strict, so the method as stated stops there. That is true as written and does not foreclose the saturation question.
[C] One more detail from the same section that the room will want and that sets up the next slide. paper/fivenomials.tex:447: "In summary, smooth rational curves of degrees 1 and 3 are reconstructed directly from their periods, whereas curves of degrees 2 and 4 are reconstructed together with their residual counterparts." The reason is at :441: a smooth rational quartic lies in exactly one quadric Q, X cap Q decomposes as C union C', and the primitive projections of [C] and [C'] differ only by a sign, so the periods cannot separate them. Slide 11 already says "predicts 168 quadrics containing residual pairs" and slide 15 counts "336 curves" against "168 pairs", both without ever saying where a residual pair comes from. One sentence here explains all three.
[C] Proposed split, with your two additions landing on the second slide:
[C] - 10a, the pairing and the theorem: the period pairing phi(gamma, omega) = int_gamma omega, the observation that (1/2.pi.i) int_gamma omega is algebraic for gamma in Pic and omega in F^1, and Movasati-Sertoz giving I(C) subsetneq I_gamma. The existing F^1 warning belongs here, since this is where the symbol reappears: it is the complex filtration, not Lecture 2's p-adic one. [C] - 10b, what can actually be reconstructed: isolated means C^2 = -2, so elliptic curves are out; degree at most 4, with the Jacobian-ideal reason; degrees 2 and 4 come with residuals; then Cifani-Pirola-Schlesinger as the concrete instance, the quadric generating I_{[C],2}.
[C] That ordering also repairs something. Today the slide states CPS for quartics immediately after a general theorem, with the hedge "In favorable circumstances we expect low order equations in I_gamma to span I(C)" carrying all the weight. Splitting lets the hypotheses be stated as hypotheses, and CPS then reads as the case where they hold rather than as a lucky special result.
[C] Bearing on an open bead, in passing. paper/fivenomials.tex:286 says "300 digits of numerical precision let to the constant B to be about 10^300 ... particularly those corresponding to smooth rational curves of degree at most 4, whose primitive projections have size less than 100", and :308 gives [C]_0^2 = -4(8 + d^2), which is -96 at d = 4. Slide 3 of this plan states the same fact in the unscaled normalisation, -(2 + d^2/4) = -6, and asks for B > 6. The two differ by the factor 16 between 4C - dH and C - (d/4)H, so they agree. minima-uv3.66 asks you to verify that the separation bound covers norm minus six; the paper states the covering claim in its own normalisation, which is most of what that bead wants.
[C] Count consequence, cumulative. With slide 9 split and slide 10 split, the proposed version is 21 spoken slides, or 22 with the title slide of minima-uv3.227, against the 45 minutes and roughly 19 slides recorded in artifacts/plan/plan.md. Dropping slide 20, the only reduction currently on the page, brings that to 20 and 21. This is now the third comment in this pass that adds a slide, so the budget is worth a decision of its own rather than being absorbed silently.
[C] Available and unused, since you are adding material for this room: paper/fivenomials.tex:88 records that "After we showed Degtyarev the 133056 example, he produced (lattice-theoretically) a deformation of Pic(X-bar) supporting a polarised K3 lattice with 258720 smooth rational quartic curves, which he conjectures to be extremal at degree 4 (private communication, 2026)." That belongs on slide 9 rather than here, but it is new, it is exactly the register of an algebraic-geometry audience, and nothing in the deck mentions it.
Deck (governs). The slide displays the period pairing, states I(C) ⊊ I_γ, keeps the favorable-circumstances hedge, and identifies I(C)₂ = I_[C],₂.
What it does. Gives the theorems that turn a Hodge class into equations for a curve.
11 Reconstructing quadric surfaces
L:L630–657
From the 133056 predicted classes, the Aut Λ orbit decomposition, and the coefficient system I_{[C],2} that determines each quadric.
Deck (governs). The slide displays the quartic and candidate-lattice inclusion, gives the full orbit decomposition, selects the orbit of 336 curves, and predicts 168 quadrics containing residual pairs.
What it does. Picks the small orbit and the quadric surfaces to reconstruct.
12 Reconstructing quadric surfaces
L:L658–692, compressed
The coefficients a_i have minimal polynomials of roughly nine thousand characters, and the same degree-168 field L is reached nine different ways.
>>[minima-uv3.251] AUTHOR (2026-09-15): another relevant communication: [Costa / Allan Steel on Magma linear algebra over number fields, saved verbatim at artifacts/picard_minicourse/_sources/magma-linalg-thread-2026.md]
[C] This supplies the real timings, and they are better than "several hours per degree" because they are one operation on one named matrix. Over the degree-168 field: M6 is 105 x 84, Rank(M6) = 74 in 1909.060 s, about 32 minutes, while EchelonForm(M6) takes 488.520 s. M7 is 196 x 120, Rank(M7) = 110 in 3364.720 s, about 56 minutes, against 925.160 s for its echelon form. Quotable as printed, since these are Magma's own time outputs.
[C] The inversion is the line for the slide, not the raw seconds. Over Q the same operations run the other way round: a random 1000 x 900 product has Rank at 0.200 s against EchelonForm at 80.730 s. So over Q rank is 400 times cheaper than echelon, and over this field it is four times dearer. Steel confirms that is a defect and not a fact of life: "Rank should always be at least as fast (and usually much faster) than EchelonForm! I will fix that in the Number Field case." That single comparison says "operations in L are seriously expensive" better than any adjective, because it shows the cost inverting a relationship the room already knows.
[C] Why it is expensive, in Steel's words, which is the honest content behind slide 14's exclamation mark. There is no modular algorithm for this input: "for high-degree fields, it is more difficult, especially if you can't easily tell whether it is going to split easily. For this input there is currently no special algorithm, so it is just using a non-modular algorithm so that's why it's slow." And the Rank-versus-Echelon gap specifically: "Rank is allowed column ops as well as row ops and I guess it must be bad luck that there is some coeff blowup in the number elts, or more inversions needed, which are very expensive here."
[C] Your own inflation idea is answered, in case it comes up from the room. You suggested reducing to linear algebra over Q by inflating with respect to a basis of K/Q; Steel: "Inflating works fine for small examples if you have no other algorithm, but the complexity is really bad and it's obviously hopeless when you have a very high degree field and large dim in the matrices, as here."
[C] THE SEVEN CLOSES THE LOOP, and this is the thing I would put on a slide above all the rest. Steel, unprompted and not knowing where the field came from: "it has a high deflation factor (i.e., p(x) = g(x^d), where d=7) and so it splits a lot mod any prime. This has caused me to come up with a faster algorithm to factor polynomials over a finite field when deflation is present, which is really good in general. But it helps a lot here because a very significant amount of time in the M6 example has been spent in just factoring p mod each prime." So the degree-168 minimal polynomial of slide 12 satisfies p(x) = g(x^7).
[C] That is the fifth independent appearance of 7 in this lecture, and it retires the speculation I logged under minima-uv3.233. The symplectic Z/7Z of the F1L3 pencil, the F_42 = AGL(1,7), the PGL(2,7), the index 343 = 7^3 on slide 18, and now the deflation factor 7 of the field slide 12 reconstructs, with 168 = 24 . 7. The corank table you sent Steel even carries a 7 in its dimension formula, (7 . C(d-4+3,3) + C(d-2+3,3)) x C(d+3,3). These are one phenomenon, and a slide can say so in a clause without proving it.
[C] It also explains the Elsenhans-Steel paper recorded two comments above. Their thread says "Allan and I got some polynomials that allow deflation from you"; this thread is where those polynomials came from and why deflation was the interesting feature. So the citation on slide 17 and the cost on slide 12 have the same origin, which is worth knowing when deciding how much of this to say.
[C] Slide 14's hedge now has a sequel, and it is a good one. Steel's closing line, after you pointed him at the Oberwolfach deck: "Working hard to try to remedy your line 'Hopeless to do this directly! Operations in L are seriously expensive!'." That is Magma's own developer reading the slide you are about to show and fixing the thing it complains about. His first modular version already brings M6's echelon form from 488 s to about 90 s, "not easy, since the result has about 3 thousands digits in numerators/denominators in the entries!".
[C] Recommendation, and the count matters because this pass has already added slides. Put the numbers on slide 12, where the degree-168 field and the nine-thousand-character minimal polynomials already are: one line with M6 or M7's rank time, and the Q comparison as the punchline. Put one clause on slide 14 noting the cost is being removed as of this year. Put the deflation seven on slide 9b, next to Aut Lambda, as the clause that ties the sevens together. None of that needs a new slide.
[C] One honest caveat before any of it is spoken. The 1909 s and 3364 s are pre-fix numbers on Steel's and your machines, and his new algorithm is already landing. So state them as what the computation cost when the work was done, not as what it costs today, or the first person to try it in current Magma will find you out.
>>[minima-uv3.237] AUTHOR (2026-09-15): Slide 12, the equation should be a bit bigger.
[C] What it is now, exactly. The polynomial is <p class="disp tiny"> and the deck's own stylesheet has .disp.tiny { font-size: 0.5em; word-break: break-all; }. So it is set at exactly half body size, and word-break: break-all means it is allowed to break in the middle of a number rather than at a term boundary. The printed string is 336 characters: four terms, x^168, x^161, x^154, x^147, with coefficients of 32, 63 and 94 digits, then a 98-digit coefficient and \cdots.
[C] Good news on scope. tiny is used exactly once in the whole course: this slide. Zero uses in talks/picard-lattices-1.html and zero in talks/picard-lattices-2.html. So the class can be changed directly and nothing else in the three decks moves.
[C] Already recorded, and this is you deciding it rather than a new finding. minima-uv3.13, "Picard: legibility spots below 60 percent of body size", logged this exact pattern on 2026-09-13 at P3, in these words: "set with class 'disp tiny' at 0.5em, half the body size, spanning the full content width". It was closed with "Folded into minima-uv3.14: one parked typography pass includes the old legibility observations". It named L1 slide 21 as the instance; that slide no longer carries the class, so this slide is the only one left in the course.
[C] PRIORITY FLAG, against your own rule. This is COSMETIC, and the plan is unsettled: six comments in this pass change what slides exist, including two splits. 00-pinned-plan-content-cosmetic records your words from earlier today, "I AM TIGHT ON TIME!! YOU KEEP WASTING MY TIME!!! PLAN > CONTENT > COMESTIC!!!", and that nothing at a lower level runs while something above it is unsettled. So this is recorded and not run. It belongs in the typography pass parked at minima-uv3.14, and it takes one line when that pass runs. Say otherwise and it goes now.
[C] The real lever here is content, not CSS, which is why this is worth a sentence rather than a number. artifacts/orch/RULES.md states: "Overflow: MEASURE and REPORT it, then LEAVE IT OVERFLOWING. Edgar cuts. Never add a font, height, clipping or wrapping rule to compensate." The string already spans the full content width at half size, so raising the font size without cutting terms overflows, and the rule forbids compensating with another font rule. Enlarging the equation therefore means printing fewer terms.
[C] That cut is already half made in the source, so it costs nothing to finish. mukai_leiden.tex:670-677 prints these four terms and \cdots and has four further terms commented out immediately below. Concretely: going from 0.5em to 0.65em is a 30 percent increase, so roughly three terms fit where four do now. Recommend dropping to the first three terms plus \cdots at 0.65em, since the point of the line is the nine-thousand-character scale and the 32 and 63 digit coefficients already make it; the 94 and 98 digit ones repeat the same impression at four times the width. Three terms at 0.65em is also above the 60 percent threshold minima-uv3.13 was measuring against, which closes that observation properly rather than leaving it parked.
[C] One thing to keep whatever size you choose. word-break: break-all is what lets a 98-digit coefficient split across lines mid-number. At a larger size with fewer terms it will still be needed, but it is worth checking on the projector that a break never lands inside a number the room is being invited to be impressed by; breaking between terms reads as a long polynomial, breaking mid-integer reads as a typesetting accident. Default taken. Compressed, per the boulders' instruction to shorten reconstruction machinery. The nine-thousand-character figure stays because it is the honest scale of the problem; the derivation goes.
Deck (governs). The visible slide gives the ten-coefficient Goal, prints the beginning of a degree-168 polynomial, and states that every computation must be selective.
What it does. Shows the cost: enormous heights and one degree-168 field presented in nine guises.
13 Isomorphism problem
L:L694–720, compressed
Building Q(a_k) → L by a Vandermonde system over compatible embeddings, with a height-reduction trick taking 120k digits down to 4k.
>>[minima-uv3.238] AUTHOR (2026-09-15): Slide 13, we need to shorten the last stencen" In practice, it is faster to iteratively refine the complex embeddings, as their height is smaller than theoretically possible: 4k vs 120k digits." so that digits does not take a second line
[C] The sentence as it stands, <p class="small">, the last visible line of the slide, 146 characters: "In practice, it is faster to iteratively refine the complex embeddings, as their height is smaller than theoretically possible: 4k vs 120k digits." Transcribed from mukai_leiden.tex:719-720, where it is the closing line of the frame and is already split across two source lines at the same place.
[C] Candidates, shortest last, all keeping "complex" and both numbers, since the 4k against 120k contrast is the whole point of the line:
[C] - 134, saves 12: "In practice it is faster to iteratively refine the complex embeddings: their height is 4k digits, not the 120k theoretically possible." [C] - 107, saves 39: "In practice it is faster to refine the complex embeddings iteratively: their height is 4k digits, not 120k." [C] - 95, saves 51: "In practice, iteratively refining the complex embeddings is faster: height 4k digits, not 120k."
[C] Recommend the middle one, at 107. Twelve characters is not a reliable margin for pulling a widow up, since the break point moves with the projector's aspect ratio, and 107 is a 27 percent cut, which clears it with room. It also fixes the limp construction: "as their height is smaller than theoretically possible: 4k vs 120k" makes the reader hold a comparison in suspense and then resolve it with a colon and a bare "vs", whereas "their height is 4k digits, not 120k" lands the same fact in one move. The 95 version reads as a telegram and drops the verb from "height 4k digits", so I would not go that far.
[C] The bigger offender on this slide is the paragraph immediately above it. That is also class="small" and it is 239 characters, well over half again as long as the one you are cutting: "Distinct nodes make {sigma_i(a_k)^j} invertible, so the solution v in Q^168 is unique; the denominators of v are bounded a priori, so enough precision pins v down exactly and the isomorphism is then verified exactly." If the slide is running long, that is where the lines are.
[C] And that paragraph is not settled, so it is worth not tuning its neighbour twice. It exists only because the panel replaced the source's claim that the system "is numerically stable, as it is a Vandermonde matrix" with the weaker true statement, and minima-uv3.8 is open on you for exactly that: supply a condition-number estimate for these nodes or a certified error bound at the precision used, and the stronger, much shorter sentence comes back. Answering minima-uv3.8 would shorten this slide more than any wording pass.
[C] Priority note, and it cuts the other way from slide 12's. This one reads as cosmetic but the fix is a wording change, which is CONTENT, so it is not parked behind the typography pass. It can be applied whenever the surrounding text is next touched.
The Vandermonde matrix {σ_i(a_k)^j} has distinct nodes, so it is invertible and the rational solution vector v ∈ Q¹⁶⁸ is unique. That is invertibility, not stability. What makes the numerical solve usable is that the denominators of v are bounded a priori, so a sufficiently precise approximation determines v exactly and the resulting isomorphism is then checked exactly. PANEL FLAG. The Leiden slide says the system "is numerically stable, as it is a Vandermonde matrix". Distinct nodes give invertibility only; Vandermonde matrices are famously ill-conditioned (Pan, arXiv:1504.02118), and compatibility of the embeddings supplies no stability guarantee. The claim above is the weaker true one. If you have a condition-number estimate for these particular nodes, or a certified error bound at the precision used, give it and the sentence can be made strong again. Keep the phrase. The slide says the isomorphism problem "feels hopeless" before the trick arrives. That reversal is the slide.
Deck (governs). The visible slide keeps that hedge and uses the weaker invertibility, bounded-denominator and exact-verification claim rather than the source's stability claim.
What it does. Solves the field-isomorphism problem through compatible embeddings and an exactly checked rational solution.
14 Intersecting the quadric surfaces with the K3 surface
L:L723–753
Goal: Q ∩ X splits into two quartics. Q is smooth, so Q_L̄ ≃ P¹ × P¹ over the algebraic closure, and Q ∩ X is a curve of bidegree (4,4) there; slide 9 already rules out components of degree 1, 2 or 3 on X, which leaves (1,3) + (3,1) and (2,2) + (2,2). The second has to be excluded separately: two nodal (2,2) curves meet in 8 points and carry a node each, so they also produce 10 reduced singular points, and the count alone does not distinguish them. What rules them out is the enumeration of Λ: a (2,2) component would be a degree-4 class of square 0, and Λ has none within the range B of slide 3, which covers primitive norm 4. That enumeration is the author's; the panel could not certify it from the supplied data, and the exclusion of (2,2) + (2,2) rests on it. With that in hand the only splitting is (1,3) + (3,1), whose two components meet in exactly 1·1 + 3·3 = 10 points, and it suffices to show the singular locus S is 10 distinct reduced points.
Smoothness buys the two rulings only over L̄: a smooth quadric over L need not be split. Nothing is lost, since the bidegree count and the component degrees are geometric statements. If the two rulings are wanted over L itself, splitness over L is a further thing to establish, not a consequence of smoothness.
[C] ASTRA RESULT, 2026-09-15, minima-uv3.234, report at artifacts/orch/lattice-astra.md. This slide's exclusion of (2,2) + (2,2) is REFUTED as an argument. Degree-four square-zero classes exist in the Picard lattice: with b1 the first basis vector of the orthogonal complement of H, b1^2 = -4, so E = H + b1 has H.E = 4 + 0 = 4 and E^2 = 4 + 0 - 4 = 0. I checked that arithmetic; it is elementary and correct given b1^2 = -4 in astra's Gram matrix. Enumeration finds 5418 such classes. Astra argues E is moreover effective, primitive and nef, a genus-one pencil whose general members are smooth elliptic quartics.
[C] What this does and does not break. Astra is careful and I am repeating its own limit: "This does not say that any particular reconstructed quadric has a (2,2)+(2,2) decomposition. It says that the proposed blanket lattice exclusion is unavailable." So the conclusion, that the splitting is (1,3) + (3,1), may well survive; the reason given for it does not. This slide currently rests the whole exclusion on that absence, and the plan says so in as many words.
[C] There is also an internal inconsistency in the deck, independent of whether astra's lattice is the right one, and it is worth stating on its own. Slide 3 asks for B > 6 so the enumeration covers every primitive norm up to 6. A degree-four square-zero class has primitive projection E - H = b1 of square -4, which is inside that range. So a Lambda satisfying the slide-3 bound must contain such classes if they exist at all, and this slide asserts it contains none. The two claims cannot both hold for the same Lambda. Astra puts it as "a claimed B > 6 bound and an asserted absence of such classes in the same Lambda cannot both hold", and calls it a consistency check rather than a computation of the bound.
[C] Bead dispositions from that run, quoted rather than paraphrased: minima-uv3.64 "Settles the claimed Picard-lattice exclusion by refutation"; minima-rza "Settles the geometric lattice calculation, with a correction"; minima-uv3.65 "Bears on it; supplies the lattice certificate", index 343; minima-uv3.63 "Bears on it; supplies a nonzero minor", 645657712; minima-uv3.66 "Does not reach it", since no certified periods or error balls were available. Nothing has been changed on any slide. This needs your verification first, because it contradicts your own computation and because astra never saw your numerical Lambda.
>>[minima-uv3.245] AUTHOR (2026-09-15): ok, I think there is a new version, that takes care of that. not sure we should focus on that right now
[C] Taken as the answer to the "it suffices" gap, and closed. The unwritten criterion at paper/fivenomials.tex:467 is written in a newer version of the paper that is not in this archive. Slide 14 keeps "It suffices to show that the singular locus $S$ of $Q \cap X$ consists of 10 distinct reduced points"; whoever drafts the slide takes the justification from the current paper rather than from the archived draft, and does not reconstruct it.
[C] Deprioritised at your instruction. Not chasing the nodal (2,2)+(2,2) case further, and not chasing astra's 5418 square-zero classes, which were only ever relevant as a way of closing that same gap. minima-uv3.64 is annotated accordingly rather than pursued.
[C] CONSEQUENCE, wider than this slide, and worth knowing once. artifacts/picard_minicourse/_sources/.../paper/fivenomials.tex is a stale draft. Several [C] notes in this pass cite it as current: the degree > 4 Jacobian-ideal argument and its commented-out status (minima-uv3.232), the two Galois-action sections and their unfinished margin notes (minima-uv3.241), the fivenomial corpus subsection and its "8 examples up to symmetry" (minima-uv3.243), the acknowledgements list (minima-uv3.236), and the residual-pairs material (minima-uv3.232). Each of those may already be settled in the version you have. None of them should be quoted to the room as the state of the work without checking against the current file. Getting that file into the archive would retire the caveat in one step.
Two halves, and they do different work. Over F_p one finds 10 distinct points, which gives only that S is zero-dimensional and reduced with deg S ≤ 10: a special fibre bounds the degree from above, it cannot prove the characteristic-zero scheme splits. The equality deg S = 10 is then certified in characteristic zero, over L itself, by the Hilbert function: one checks dim L[x,y,z,w]_m / I_m = 10 for m = 6, 7, where V(I) = S.
>>[minima-uv3.239] AUTHOR (2026-09-15): we ar emissing the emojis in slide 14
[C] Confirmed, and it is deck-wide rather than local: talks/picard-lattices-1.html, -2 and -3 contain zero emoji characters between them. The Leiden source has exactly three, and all three were dropped in transcription.
[C] The full inventory, with locators. mukai_leiden.tex:687, \emoji{fearful-face}, in the frame starting at 658, attached to "The abstract isomorphism problem feels hopeless." mukai_leiden.tex:736, \emoji{anxious-face-with-sweat}, in the frame starting at 723, attached to "Linear algebra". mukai_leiden.tex:738, \emoji{face-screaming-in-fear}, same frame, attached to "Gröbner basis".
[C] So it is one running joke over two slides, and the deck broke it in the middle. The first emoji now belongs to slide 13, because the deck moved "The abstract isomorphism problem feels hopeless" out of the slide-12 frame into slide 13's hedge box, where it currently sits bare. The other two are the escalation on slide 14. Restoring only slide 14 gives you the punchline without the setup.
[C] What the transcription did to the line, which is worse than dropping a decoration. The source reads "Linear algebra \emoji{anxious-face-with-sweat}" then "Gröbner basis \emoji{face-screaming-in-fear}", with a \pause between them, so the room sees the cost get worse one step at a time. The deck renders that whole thing as five words: "Linear algebra. Gröbner basis." Two noun phrases and two full stops, with nothing saying why they are there. Without the emoji the line is not a weaker joke, it is not a sentence.
[C] The staging went too, and it is part of the same edit. Slide 14 has no fragments at all, so the three \pause steps in this frame, before "Hopeless to do this directly", between the two emoji lines, and before the F_p bullet, are gone and the whole slide arrives at once. Restoring the emoji without the pause gives both faces simultaneously, which is the joke told backwards. Whether the pauses come back is pacing and yours, but the two questions are the same question.
[C] One build check before anyone applies this. artifacts/publish/deck-build/build.py verifies the output for a URL allowlist, inline CSS, forbidden math libraries and raw LaTeX; emoji are ordinary Unicode and none of those gates touches them. The real risk is rendering, not building: the deck loads only Source Sans Pro from Google Fonts, so an emoji falls through to whatever the projector machine has. Worth one look at the actual venue rather than assuming, since a tofu box would be worse than the full stops.
>>[minima-uv3.240] AUTHOR (2026-09-15): "We conclude ..." is weirdly broken in2 lines
[C] Same slide, last bullet, and the cause is measurable. The bullet is 180 characters, the longest of the four on the slide, and it carries an unbreakable math span 44 characters wide: $\dim L[x,y,z,w]_{\bullet}/I_{\bullet} = 10$. Inline math cannot wrap, so the line has to break before it, which is what leaves the short first line and the odd second one. The other three bullets are 101, 173 and 121 characters and none has a span anywhere near that width.
[C] There is also a straight typographic clash, which may be the "weirdly" you are seeing. The bullet is an item in <ul class="tight">, and the math inside it uses \bullet three times as the subscript placeholder, so the line renders bullet glyphs inside a bulleted item. Two different meanings for the same dot, one inside the other.
[C] Your own plan already fixes it. The written entry for this slide says "one checks dim L[x,y,z,w]_m / I_m = 10 for m = 6, 7", using m where the deck uses \bullet. Adopting the plan's m shortens the unbreakable span from 44 characters to about 30, removes the glyph clash, and needs no rewording. That is the smallest change that addresses both halves of what you noticed.
[C] If it still breaks badly after that, the next cut is the saturation clause. "where $I$ is saturated" is the panel's addition and is not in the source line; it is load-bearing, since Gotzmann needs it, but it is also the natural thing to move into the speaker notes, where the reason it is needed is already written out.
[C] Note for whoever applies this: \bullet is used only inside this one bullet on this slide, so replacing it with m is local and cannot disturb notation elsewhere in the lecture.
Gotzmann regularity is what licenses stopping at finitely many degrees, and it needs its hypothesis said out loud: it bounds the regularity of a saturated ideal in terms of its Hilbert polynomial, so the ideal one computes with has to be saturated (or its regularity certified some other way). Without that, adding a degree-8 equation can leave the values in degrees 6 and 7 untouched while shrinking V(I). Once I is known to be saturated, the single value h_I(6) = 10 together with deg S ≤ 10 from the special fibre already forces deg S = 10, and the scheme-theoretic statement follows rather than being sampled.
Deck (governs). The visible slide states the 10-point criterion, the F_p computation, saturation and the degree-6,7 Hilbert-function check; the bidegree exclusions and division of labour are in the speaker notes.
What it does. The candidates become curves. This is the moment the numerics turn into geometry.
15 Certifying Pic X̄ = Λ
L:L755–791
Λ_Q := ⟨[C] : C ⊂ σ(Q) ∩ X⟩ ⊆ Pic(X̄)|_B ⊆ Λ ⊆? Pic X̄
Verbatim, including: "The inclusion Λ_Q ⊆ Λ is not explicit!" Then the rescue: both Pic X̄ and Λ are saturated in H₂(X,Z), so it suffices that rank Λ_Q = rank Λ = 19.
The rank equality is half of it. The other half is the upper bound rank Pic X̄ ≤ 19 from slide 9: saturation alone would leave Picard rank 20 open, with Λ a corank-one saturated sublattice. With the upper bound in hand, Λ_Q ⊆ Λ and Λ_Q ⊆ Pic X̄ all have rank 19, so Λ and Pic X̄ are saturated lattices with the same rational span, hence equal.
Two routes to the rank equality: intersect the 336 curves with each other over F_p, or certify that the quadrics correspond to the classes they are supposed to. The second route counts twice over, and the two counts are at different scales. Globally, at most 66528 quadrics arise from the 133056 predicted quartic classes, one per residual pair, and this can be done over C. Locally, it is the 168 conjugates σ(Q), one for each embedding σ : L → C, that are in bijection with the 168 pairs of quartic classes cut out by them, accounting for the 336 curves. The bijection is between those two sets of size 168, not between the 66528 and the 168. Either route still owes the rank itself: an explicit certificate that rank Λ_Q = 19, a nonzero 19 × 19 minor of the intersection matrix of the 336 classes, or a prior verification that those 336 vectors span Λ. With that in hand, Pic X̄ = Λ, with the certificate mark. 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 carried out. It is stated above as a requirement, not as done.
Deck (governs). The visible slide states the saturation rescue and concludes Pic X̄ = Λ; its speaker notes retain the missing rank-19 certificate as a requirement.
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.
16 Computing the Galois action
L:L794–819
Q ∩ X splits into a pair of quartics over K, a quadratic extension of L. Goal: compute K and Gal(K/Q) acting on Λ_Q.
>>[minima-uv3.249] AUTHOR (2026-09-15): For Galois see: [forwarded conversation "Galois group", Elsenhans / Sertoz / Costa, 10-19 August 2025, saved verbatim at artifacts/picard_minicourse/_sources/galois-thread-2025-08.md]
[C] This is the answer to the standing question on this slide, and it says the slide is aimed at the wrong target. Of the three closing questions, the first is essentially done and the other two are not independent of it.
[C] Computing the group is not hopeless. Elsenhans, 10 August 2025: "I had a look at the degree 96 examples. Magma can compute all the groups in about 1 hour. Most of the time is used to find the subfields. The attached file computes the groups in about 5 minutes." The two difficulties he names are algorithm selection in the normalizer for permutation groups, and "a very long descent chain of index 2 and index 4 subgroups".
[C] Found is not proven, and the slide should not blur them. His prescription, verbatim: "Use the Galois groups found to compute proven subfield data. Start the Galois program once more with the full subfield information. Similar to the degree 2*168 example, this should result in proven groups."
[C] What IS hopeless-looking is the transfer, and the slide never mentions it. Your own message of 18 August: "We were wondering if after the fact one can convert the Galois group action on the p-adic roots to the complex roots. I tried to modify the code that you sent us but I quickly realized that trying to force it to use Type:="Complex" would not work." And the reason it matters, also yours: "If we would have its representation as the permutation group on the complex roots, then we have it as a permutation acting on the Galois orbit of vectors inside of the picard group, and thus we could pull it back into the automorphism group of the Picard lattice. Without that we do not really have the Galois action on these orbits."
[C] Sertoz states the constraint that makes the transfer unavoidable rather than a convenience, 19 August: "We identify them with the complex embeddings of a number field K ... where the vi are in bijection with the complex roots of p(x). This identification is non-negotiable: we work with periods of the K3 to deduce the complex embeddings first, and only then reconstruct K." And why the obvious route fails: passing to the Galois closure and writing each root over a primitive element "would in theory be transferable between complex embeddings and p-adics. But in practice, the Galois fields that appear are enormous, and this approach looks unworkable."
[C] So the second and third closing questions are not three independent open problems. "Is it Aut Lambda" and "what is H1(Gal, Pic X-bar)" both need the action on the orbit of vectors in the Picard group, which is exactly what the complex identification supplies. They are blocked behind the transfer, not beside it.
[C] The running example is one of these fields, which is what makes the thread quotable on this slide rather than merely relevant. Elsenhans singles out "the degree 2*168 example" and then "the degree 168 / 336 example" twice, and this slide's field has [K:Q] = 336 = 2 . 168. His diagnosis of why they are hard is one clause: "all these examples have in common that the stem field is close to the splitting field. In some sence this is the worst case for the GaloisGroup program."
[C] Recommended reshape, which is yours to accept or refuse. Replace "The direct computation of Gal(K/Q) looks hopeless" with the true division: the group is computable in minutes; the periods fix an identification with the complex roots that the p-adic computation cannot see; bridging the two is open. That converts a shrug into one sharp problem handed to the room, and Sertoz's closing line is the invitation already written: "So we need something more clever - perhaps involving finite fields? Maybe we're overlooking something obvious."
[C] DATE CAVEAT. The thread is August 2025, thirteen months before the lecture. The current paper carries a "Galois action, revised" section and minima-uv3.247 and minima-uv3.248 are checking what has moved. Nothing above should reach the room as today's status without that check.
>>[minima-uv3.250] AUTHOR (2026-09-15): ok, we should say in Slide 17, there is a new paper about computing Galois groups of this kind of polynomial [Elsenhans--Steel] (as in Allan Steel)
[C] Visible form is settled by an existing rule, so the slide can be written now. The deck-wide convention from minima-uv3.96 is that visible citations name authors only, with no year and no theorem number, so [Elsenhans-Steel] is already the correct thing to print. Only the full reference in the speaker notes is outstanding.
[C] RESOLVED, and it is not on arXiv because it does not exist yet. Second thread, August 2026, now saved with the first at artifacts/picard_minicourse/_sources/galois-thread-2025-08.md. Elsenhans: "Some time ago, Allan and I got some polynomials that allow deflation from you. We are working on a paper describing the corresponding algorithms. Do you have something that we can cite? Our paper would look better if we could cite an application." So my searches were correct to come up empty; the paper is in preparation.
[C] The technical word is DEFLATION, not "Galois groups of this kind of polynomial". That is what Elsenhans and Steel's algorithms do, and it is consistent with the first thread's diagnosis: the cost was all in finding subfields, and deflation is what reduces that. Worth using the right word on the slide, since it is the one the eventual citation will carry.
[C] Visible form. [Elsenhans-Steel] is correct under the authors-only rule of minima-uv3.96, with "in preparation" spoken rather than printed. Do not print a year; there is not one yet.
[C] The reciprocity is a live obligation, not just context. They want to cite YOUR work as the application. Sertoz's reply: your own paper is "sitting at 95% complete", he cannot finish in September, "mid-to-late October would be plausible", and he offers them the working title "Rigorous reconstruction of the Picard lattice as a Galois module from periods of quartic K3s" to cite as a work in progress. So there is a coordination question outside this lecture, and a deadline in it.
[C] That working title is usable in the deck today, which nothing in the plan currently exploits. It names exactly what Lecture 3 does, in one line, in your own words, and the course has no self-citation anywhere. Candidate homes: slide 18, which states the theorem and then the "Wanna be a Theorem", and slide 19, where minima-uv3.243 is already adding the five-nomial scope. A line reading that the method is written up as "Rigorous reconstruction of the Picard lattice as a Galois module from periods of quartic K3s", in preparation with Sertoz, tells the room what to look for and when.
[C] Timing consequence, and it is the one that could embarrass the slide. Your paper is unposted and due mid-to-late October; theirs is unposted and waiting on yours. So on 16 September neither citation resolves to anything a listener can fetch. Both have to be spoken as in preparation. If either lands before the deck is posted publicly, the slide needs the handle added rather than left as a bare bracket.
[C] What I did find, so nobody wastes the search again. Andreas-Stephan Elsenhans, "Computation of Galois Groups in magma", in the ICMS proceedings (Springer), a degree-independent algorithm as implemented from Magma 2.21. Elsenhans-Kluners, "Computing subfields of number fields and applications to Galois group computations", arXiv:1610.06837, which is directly the bottleneck the thread names, since Elsenhans writes that "Most of the time is used to find the subfields". And the Magma handbook records that for polynomials over Q, Elsenhans has a method for proving correctness that can be run on the result of the GaloisGroup intrinsic, which is the found-versus-proven distinction from the thread.
[C] Allan Steel is corroborated as involved: he is named in the paper's acknowledgements, and the August 2025 thread has Elsenhans saying "Make sure you use the latest version in svn", which is Magma's own repository. So the credit shape is right even though I cannot cite the paper.
[C] Where it goes on the slide. This is the citation for the claim the previous comment reshapes: the group computation is feasible, and now it is feasible in published work rather than in a colleague's attached file. That is a materially stronger line than the thread alone supports, and it is the one that makes the open problem clean, since the room can be pointed at a paper for the part that works and at the p-adic-to-complex transfer for the part that does not.
[C] Consequence for the hedge. With a published method cited, "The direct computation of Gal(K/Q) looks hopeless" cannot stay on the slide as written; it would be contradicted two lines later by its own citation.
>>[minima-uv3.241] AUTHOR (2026-09-15): the paper already answered "Can one compute KK using geometry without Gröbner basis?" in Slide 16, no?
[C] Yes, and with a closed form rather than a procedure. paper/fivenomials.tex:481, in the section "Some projective geometry", subsection "Residual quartics in a quadric surface", verbatim: "Lemma. Suppose $X \cap Q$ is the union of two curves $C' \cup C''$ each of arithmetic genus $0$ and degree $4$. Then the curves $C'$ and $C''$ are defined over $K' = K(\sqrt{\disc(q)})$." Here $q$ is the quadric and $\disc(q)$ its discriminant as a quadratic form. That is exactly the $K$ this slide is asking for, read straight off the ten coefficients $a_i$ that slide 12 reconstructs. No Gröbner basis, no hyperplane, no degree-8 scheme, no orbit count.
[C] The proof is four lines of classical geometry, which is what makes it slide material rather than a citation. The two rulings of a smooth quadric are defined over $K(\sqrt{\disc q})$; over the closure $Q \simeq P^1 \times P^1$ and $C' \cup C''$ has bidegree $(4,4)$; arithmetic genus 0 forces $(1,3)$ and $(3,1)$ by degree-genus; and the nontrivial element of $\Gal(K'/K)$ swaps the rulings, hence swaps those two classes, so the components are individually defined exactly over $K'$. Slide 14 already does the first three of those steps in its speaker notes, so the slide is most of the way there without knowing it.
[C] So two lines on the slide are now stale, and they are the last two. "Can one compute $K$ using geometry without Gröbner basis?" is answered yes and should become the Lemma. "To try: For a generic hyperplane $Q \cap X \cap H$ is a degree 8 reduced scheme. The number field $K$ is the quadratic extension where we observe two orbits." was the proposal that the Lemma replaces; it is strictly worse, since it needs a generic hyperplane, a degree-8 computation and an orbit count where the Lemma needs one discriminant.
[C] Gröbner is genuinely gone from this route, not merely avoided on this slide. "Gröbner" occurs exactly once in the whole 663-line paper, at :431, and it is about lines: "it is more straightforward to construct the equations of the lines in $X$ algebraically using Gröbner bases". Nothing in the field-of-definition argument uses one.
[C] What is NOT answered, and the slide should keep. The Goal box asks for two things, "Compute $K$ and $\Gal(K/\mathbf{Q})$ acting on $\Lambda_Q$". The Lemma gives the first and says nothing about the second. So the slide keeps its Goal, loses its question, and the open half moves cleanly onto slide 17, which is where it belongs.
[C] The other hedge on the slide is partly dissolved rather than answered, which is worth being precise about. "Unclear how to certify this step! What are the denominators of $\frac{1}{2\pi i}\int_C \omega$?" is about reconstructing the period coordinates in $K^{21}$. The Lemma reaches $K$ without reconstructing any periods, so for the purpose of computing $K$ that worry no longer applies. It has not been answered in its own terms, and it may still bite for the Galois action, so recommend keeping the hedge but attaching it to what still needs the periods rather than to $K$.
[C] One hypothesis to check before it is stated as a theorem. The Lemma assumes $Q$ smooth, and the paper's own margin note immediately below it reads: "I think I proved at some point that if $C'$ and $C''$ are smooth (or even only one) then $Q$ must be smooth. You may have to mention that singular Q never appears so the lemma above is 'complete'." So the completeness of the smoothness hypothesis is an open editorial point in the draft. Slide 14 already asserts "Q is smooth" as part of its bidegree argument, so the two slides need the same justification and should get it once.
[C] Consequence for slide 17, which is larger than this comment. Slide 17 ends on three open questions, two of which are "can we compute $\Gal(K/\mathbf{Q})$" and "is it $\Aut \Lambda$", under the verbatim line "The direct computation of $\Gal(K/\mathbf{Q})$ looks hopeless." The paper now has two sections aimed at exactly that, "Galois action" at :561 and "Galois action, revised" at :615, which compute the action on $\Lambda_{\le 4}$ by pulling back the known permutation actions on Galois orbits through $O(\Lambda_{\le 4}, h_X) \hookrightarrow \prod_d O(\Lambda_d, h_X)$, precisely to avoid constructing the splitting field. That is not finished: the draft carries "ooops, this is a power of 2, and I'm not sure what it has to be", "currently, we need to be smart about this", and "Good news: It looks like we will be able to skip the previous step". So slide 17's questions are not stale the way this one is, but they are no longer untouched either, and you are the only person who knows how far that section has moved since.
[C] A detail from those sections that corroborates this slide's framing, for free. :565 says the cases $d = 2$ and $d = 4$ are "more delicate: here the reconstructed classes occur in residual pairs, and only the equivalence classes of primitive projections (up to sign) are defined over a subfield $\overline K_d \subset K_d$ of index 2." That index 2 is this slide's "quadratic extension of $L$", arrived at independently, and it is the same phenomenon slide 10's residual pairs produce. Three slides, one fact. Verbatim: "Unclear how to certify this step! What are the denominators of (1/2πi) ∫_C ω?" And the proposal to try geometry instead: for a generic hyperplane, Q ∩ X ∩ H is a degree-8 reduced scheme, and K is where two orbits appear.
Deck (governs). The slide keeps the uncertainty visible, displays the period coordinates in K²¹, and asks whether geometry can compute K without a Gröbner basis.
What it does. Poses the Galois-action problem and the certification gap before offering the geometric route.
17 Computing the Galois action
L:L821–849
Verbatim: "The direct computation of Gal(K/Q) looks hopeless." Then: "We guess that K = F(u^(1/14))" with [F:Q] = 24 and Gal(F/Q) = C₃ × PGL(2,7), noting #Gal(F/Q) is 14 times smaller than #Aut Pic X̄. Ending on three open questions: can we compute Gal(K/Q); is it Aut Λ; what is H¹(Gal, Pic X̄).
The degrees now close up, and it is worth saying so out loud: [K:Q] = [K:F][F:Q] = 14 · 24 = 336 = 2 · 168 = [K:L][L:Q], matching slide 16's quadratic extension of the degree-168 field L of slide 12. PANEL FLAG. One number on this slide is still not self-consistent as written. [F:Q] = 24 but C₃ × PGL(2,7) has order 1008, so F cannot be Galois over Q with that group. The reading that makes the whole slide work is that Gal(F/Q) denotes the Galois group of the Galois closure of F/Q, of order 1008, since 1008 · 14 = 14112 = #Aut Pic X̄ is exactly the "14 times smaller" in the hedge. Confirm that reading before speaking to it.
What it does. Under the purpose you named, this is not an apology for an incomplete result. It is three questions handed to a room full of people who could answer them.
18 Summary
L:L851–870
The quartic X : x⁴ + xyzw + y³z + yw³ + z³w = 0 has Pic X̄ equal to the saturation of the lattice generated by the 336 Galois-conjugate smooth rational quartics, all of them defined over the normal closure of K/Q.
Two words in that statement are doing work that the short form hides. A quadratic extension of L = Q({a_i}) splits one quadric section into its two components, and that is all it does; the field over which every conjugate curve is simultaneously defined, and so the field the Galois action is computed over, is the normal closure, not K. And the 336 curves generate only a sublattice of index 343 in Pic X̄, a value asserted from the author's computation and not independently checkable by the panel, so the equality is with the saturation and the saturation step is not cosmetic. That is also why the wanna-be theorem below asks for exactly that algorithm. Verbatim: "We are still developing the method", and the saturation algorithm stated as a "Wanna be a Theorem (C-Sertöz)": a practical algorithm to compute the saturation of the lattice generated by rational curves of degree up to 4.
Deck (governs). The visible theorem instead says Pic X̄ = Λ, generated by quartics over a quadratic extension of L; the saturation and normal-closure correction remains an unapproved candidate in the speaker notes.
What it does. States the theorem the lecture proves for the running example, then leaves the practical saturation algorithm unanswered.
19 Do you have a challenge K3 surface for us?
L:L868
The last line of the Leiden deck, and the last line of the course.
>>[minima-uv3.255] AUTHOR (2026-09-15): and for the last slide we should try to summarize the computation so far, in particular, the example that we are stuck on
[C] This changes what slide 19 is. It currently holds one centered line, "Do you have a challenge K3 surface for us?", and minima-uv3.243 already adds the five-nomial scope to it. With a status summary and a named stuck example it becomes a real closing slide rather than a bare invitation, and the ask gets teeth: not "give us a hard surface" but "here is exactly where we stop, can you get past it".
[C] The data is being gathered now, not guessed. minima-uv3.248 has astra reading the 117 status notes in five-nomial-current/docs/superpowers/ for precisely this: which surfaces are finished, which are in progress, which are stuck, as a table of label, rank and state. The newest note is dated 2026-08-18, so it is more current than the paper.
[C] Candidates for the stuck example, from the file names alone and NOT to be used until astra confirms: 2026-08-03-x156-path-alternatives.md and 2026-08-02-anchor-feasibility.md point at X156; 2026-05-13-x110-inclusion-handoff.md and the X110_refine15000_2026-07-29 and X110_p36000_diag.log artefacts point at X110; X92_p300probe_rescue1.log at X92. There are also per-surface verification notes for M1, M3, M4, M5, M6, M7 and X13, which reads like the finished set. 2026-08-03-eta-and-abort-criteria.md suggests you already have written criteria for when to give up on one, which would be the sharpest thing to show.
[C] Consequence for the slide's own purpose, which is worth deciding with it. Slide 19's notes say to leave it up during questions, and minima-uv3.242 made it the last slide by dropping 20. A summary table plus a stuck example plus the scope line plus the question is four things on a slide that currently has one. Either it stops being the leave-it-up slide, or the summary goes on a new penultimate slide and 19 stays bare. That is a structural choice and it is yours; my recommendation is the new penultimate slide, since it also gives the Magma and Sage oral lines somewhere to attach.
>>[minima-uv3.256] AUTHOR (2026-09-15): that is much worse than mukai
[C] Ambiguous antecedent, so recorded and flagged rather than applied. It followed my list of candidate stuck examples, X156, X110 and X92, so it reads as "the stuck example is much harder than the Mukai quartic". Which one is not determined by the sentence, and the notes give three materially different pathologies, so I have not guessed.
[C] astra's findings, from minima-uv3.248, with the labels the notes use. Your running example is X153 in that labelling. The corpus has exactly 161 S4-orbits.
[C] - X110, rank 17. Stuck on a labelling bug, and the symptom is the most dramatic on the list: "560 of 576 positions mismatch". The July 31 note supersedes the earlier precision and denominator diagnosis, and its fix is proposed rather than completed. It also carries the general lesson "no per-coefficient heuristic can be both safe and complete", and that the exact verification "caught a false positive". [C] - X156. Stuck or in progress as of 7 August: "both still in their final family", with two symbolic runs carrying 9 August timeout deadlines and no completion or timeout outcome recorded. [C] - X141, X142, X157, X175, all rank 20. Partial or stuck in the primitive-element search, verified for X141 and assumed for the other three. These are worse than Mukai by rank rather than by bug: 20 is maximal for a complex K3, against X153's 19. [C] - X97, X108, X121, X137, X138, X159, all rank 18. Partial or stuck on a KeyError, diagnosed for X97 and assumed for the rest.
[C] If "much worse" means harder mathematics, it is the rank-20 four. If it means the computation went worst, it is X110. Tell me which and the slide gets the right one; both make a good closing slide for different reasons, X110 because the failure is concrete and diagnosed, the rank-20 set because it is the edge of the method.
>>[minima-uv3.257] AUTHOR (2026-09-15): there is one example, I think X110 where we are trying to compute the periods to some incredibly high rpecison
[C] Confirmed, and the numbers are in the notes. X110 is classed moreprec rather than ok, and the figures are: degree 576, a field K of degree 288, and periods at 36000 bits, which is the precision that appears 21 times across the notes and is the working setting. 2026-05-13-inclusion-per-coefficient-denominator-bound.md:30 puts it as "X110 scale (deg 576, prec 43 kbits, K of deg 288)". code/finished/X110_p36000.sobj is the phase 2/3 output at 36000 bits, dated 8 May.
[C] It was pushed further than that, and this is the part that makes it the stuck example rather than a slow one. 2026-07-26-x110-highprec-integration-failure-handoff.md is titled "X110 high-precision period computation is failing (not slow)", and records "A debug.py run has been grinding X110 at 72000 bits on lovelace since". So the honest state is not that 36000 bits takes too long; it is that the computation fails at the precision the surface seems to need.
[C] The comparison with the running example is the line for the slide, and it needs no rhetoric. Slide 3 says the working precision for X153 is 300 digits, giving B about 10^300. X110 needs 36000 bits, which is about 10800 digits, and was attempted at 72000 bits, about 21700 digits. So the stuck example sits at roughly thirty-six to seventy-two times the precision of the one the whole lecture is built on. That single ratio says more about the frontier than any adjective.
[C] Supporting figures if a second line is wanted. The full ten-row X110 pipeline at 36000 bits was projected at about 24 hours sequential, with a parallel target at 32 workers. Low-precision probes at nbits=300 on X5, X41, X54 and X110 each ran 20 to 50 minutes, so even the cheap end is not cheap. And the labelling bug recorded on 31 July gives the other half of the failure: "560 of 576 positions mismatch", with its fix proposed rather than completed.
[C] So the answer to your earlier "much worse than mukai" is X110, and the reason is precision rather than rank. It is rank 17, LOWER than X153's 19, which is the counterintuitive part worth saying out loud: the obstruction is not how big the Picard lattice is but how much precision the periods demand, and X110 demands two orders of magnitude more digits than the example that produced 133056 curves.
[C] Recommended shape for the closing summary, drawing on minima-uv3.245 and this. Three lines: the corpus, 161 S4-orbits of smooth unit-coefficient five-nomial quartics, being worked through; the method's reach, X153 at 300 digits giving rank 19 and 133056 smooth rational quartics; and the wall, X110 at rank 17 needing about 10800 digits and failing at 21700. Then the existing question. That is a status summary in three lines rather than a table, which keeps slide 19 readable if you decide to put it there rather than on a new penultimate slide.
[C] CORRECTION (2026-09-15, from minima-uv3.259). Three claims I made above about X110 are wrong, and this block supersedes them. Do not draft from the earlier wording.
[C] - WRONG: that X110 "needs 36000 bits, about 10800 digits, and was attempted at 72000 bits", giving "thirty-six to seventy-two times the precision" of the running example. The 72000-bit run is a string of upstream integration failures, "Failed to integrate fragment ... Try increasing nbits", not a demonstrated precision requirement. The real inclusion tiers are 43227, 86454 and 172908 bits, with reconstruction failing across those doublings and six configurations, and an older log stalling at 345816 bits inside a Vandermonde inversion. [C] - WRONG: that X110_refine15000 is a bit count. It is 15000 requested DECIMAL digits, and it is a completed period artifact, 19.6 hours wall and 199.7 CPU-hours, never banked. [C] - MISLEADING: treating the census's 997 bits as a working precision. It is the legacy database parent precision, which is exactly what your open bead fcv flags as a roughly 115-digit over-trust. It is not a measured accuracy and should not be quoted as one.
[C] The failing step is now named, and it is not precision at all. The obstruction is compatible field inclusion and rational reconstruction: L.inclusion(K, ordered_embeddings), embedding the degree-288 coefficient field into a degree-576 primitive field. The 31 July diagnosis is that the saved per-row match array, Kindex, assigns the wrong conjugate-to-embedding labels: row 9 forms exactly 288 pairs, but the saved labels agree at only 16 positions. That supersedes the earlier "enormous denominators" explanation, which the note explicitly retires.
[C] So the honest closing line is not "X110 needs absurd precision". It is that the periods are computed and the step that fails is matching conjugates to embeddings. That is the SAME obstruction as the Galois correspondence saved in _sources/galois-thread-2025-08.md, where the p-adic Galois action cannot be transported to the complex roots because the period-derived identification is the thing that has to be matched. One obstruction, surfacing in two places, which is a considerably better closing slide than two unrelated difficulties.
[C] Corpus count settled at 161, and the 189 is not duplication. Your own tracker decides it: b5j holds the equation audit and closed x66 the completed enumeration. The 189 legacy INDEX rows are 154 corpus representatives, 10 four-term equations, 22 six-term equations and 3 five-term equations with non-unit coefficients; the corpus is those 154 plus M1 to M7, giving 161. X31, X176 and X183 carry coefficients -8, -1 and -9 and are S4-equivalent to the genuine unit-coefficient X156. Bead m0v does NOT explain the gap; it concerns filename-based coverage, not membership. Consequence: my count of "125 of 189 at 997 bits" above has the wrong denominator and should be recomputed over the 161 before any fraction is spoken.
[C] Slide 17 keeps its hedge, per minima-uv3.247. The 2 August paper leaves both Galois sections unchanged and still visibly unfinished, so "The direct computation of Gal(K/Q) looks hopeless" is NOT superseded. Per question: computing Gal(K/Q) is not established, and the paper says that if the kernel is larger "additional work is required", possibly constructing the full splitting field; equality with Aut Lambda is neither established nor stated for the example, the revised section ending only at G' subset G; and H1(Gal, Pic X-bar) has the value 0 asserted in the rough Mukai data as Br_1(X)/Br(K) = H^1(G_K, Z^rho) = 0, while the stated theorem still reads "The algebraic part of the Brauer group is [fill in]" and the Brauer subsection remains a TODO. Do not present 0 as a finished theorem, and do not present the question as wholly untouched. One trap: the paper records # G_K = 14112 without identifying that notation with this slide's curve field K, giving a structure, or proving equality with Aut Lambda.
[C] And slide 14's gap is still open against the version I can read. minima-uv3.247 reports "no singular-locus criterion was added": the TODO at the head of the projective-geometry section is unchanged in the 2 August paper. You said a newer version takes care of it, so either your working copy is later than 2 August or that gap is live. Nothing to do about it here, but it should not be assumed closed.
[C] SECOND CORRECTION (2026-09-15). Two agents on two machines answered, and THEY CONTRADICT EACH OTHER ON X110. Nothing about the X110 diagnosis should reach a slide until you settle it. My correction block above, which said the labelling bug supersedes the denominators explanation, is now itself disputed.
[C] Reply 1, https://gist.github.com/edgarcosta/3b7dec8b6644d1ca8731c3c86f5400e6, from the machine that HAS the July and August artifacts: the failing step is NumberField.relative_quadratic_model, recovering t and n in K via element_from_conjugates; candidates at 86454 and 172908 bits have denominators of about prec/2 and are rejected by mod-p screening; root cause identified but UNTESTED, namely that the saved Kindex matches row-9 data at only 16 of 576 positions while the L-side matches 576/576, so a permuted pairing makes t fall outside K. Its own status line: "OPEN, mechanism unidentified".
[C] Reply 2, https://gist.github.com/edgarcosta/097693c4d066bfc08a11c9a422401733, from a machine four months stale, newest commit 2026-05-14, which does NOT have the July 31 handoff, the 47 probe files, X110_refine15000 or the 72000-bit run. It makes two provenance corrections that are claims about where a string comes from rather than about the data: "560 of 576 positions mismatch" is a PROGRESS COUNTER emitted during Vandermonde construction, not a mismatch statistic; and "72000 bits" is an X15 deformation-chain step counter, not an X110 result. Its diagnosis: the degree-288 polynomial WAS recovered at 43227 bits and verified irreducible, the failing call is L.inclusion(K, ordered_embeddings), and the 576 per-coefficient denominators are PAIRWISE COPRIME at about prec/2 bits each, so their LCM outruns the precision at every doubling, with timings up about 8x per doubling and an extrapolation of months to years. It classifies this as a precision and height wall, explicitly not a labelling bug.
[C] They agree on WHERE and disagree on WHY. Both put the failure in the inclusion chain, and both measure denominators at about prec/2. One says fix the conjugate-to-embedding labels; the other says the heights are genuinely out of reach and the labelling figure was a misread counter. Note the stale machine's corrections do not require the missing artifacts, since they are about which code emits the string, so being stale does not by itself settle it against them.
[C] FOR THE SLIDE TONIGHT, the safe form is what both agree on and nothing more: the periods are computed, the degree-288 polynomial is recovered, and the step that fails is the compatible field inclusion, where the coefficient denominators run at about half the working precision. That sentence is true under either diagnosis. Do NOT say "labelling bug" and do NOT say "needs 72000 bits".
[C] Also retracted from my earlier notes, twice over: the "36 to 72 times the precision" framing, and the claim that the labelling bug supersedes the denominators. The first rests on a step counter, the second is contested.
[C] Field degrees, reply 2 refining reply 1, and it kills the column outright: there is no single degree per surface, because 21 of the 22 surfaces with known degrees carry MORE THAN ONE field. Only 22 of 189 have real computed degrees. Orbit size bounds the degree but does not equal it; the degree divides the orbit size by a ratio that varies per surface. The clean anchor is X153: a 336-curve orbit reducing to 168 distinct Grassmannian columns, field degree 168. So the closing slide can carry that one anchor and must not print a corpus-wide range.
[C] Lattice, bearing on minima-uv3.270. Reply 1 supplies real invariants: det Lambda = 1372 = 2^2 . 7^3, signature (1,18), H-perp of determinant 7^3, discriminant group Z/7 x Z/7 x Z/28, Aut generators of orders 21, 6, 2, 8. Reply 2 adds the observation that matters for slide 9: BOTH candidate structures have order 14112, so the order agreeing proves nothing about which group it is, and neither machine computed the structure. The deck's existing hedge, that Aut Lambda "seems to be" F_42 x PGL(2,7), is therefore exactly right as it stands and should not be strengthened tonight.
>>[minima-uv3.258] AUTHOR (2026-09-15): the issue, is that we are trying to recover a polynomial of degree 288, which we can deduce via Galois theory, but then we cannot finish it off somehow. let's just provide a range of "examples", like most work with every little precision, we should try to clarify what is size of the number field
[C] The range exists and is countable. code/INDEX.md is the census, 189 rows, one per orbit representative, with columns for the equation, a precision in bits, the Picard rank, probe artefacts and a status tag. Three precision tiers only: 997 bits for 125 surfaces, 4120 bits for 44, and 337 bits for 20. Status: ok for 112, notfull for 48, moreprec for 24, wrong for 5. Ranks run the whole way from 1 to 20, with rank 4 the commonest at 40 surfaces, then rank 10 at 29 and rank 18 at 22.
[C] So "most work with very little precision" is exactly right and now has a number: 125 of 189 sit at 997 bits, which is about 300 digits, the same working precision slide 3 quotes for the running example. Two thirds of the corpus needs nothing beyond what the lecture already describes.
[C] The outlier is the slide. X110 carries the same 997-bit default in the census, is tagged moreprec, and then appears with 47 separate X110_300probe_* artefacts and an X110_refine15000, before the 36000-bit and 72000-bit runs. So the story is not that X110 was always expensive; it is that it starts in the ordinary tier and then needs two orders of magnitude more precision than anything else in the corpus.
[C] WHAT YOU ASKED FOR IS NOT IN THE CENSUS. INDEX.md records precision, rank and status; it does NOT record the degree of the number field. The only field degrees I can source are per-surface prose: K of degree 288 for X110, and the degree-168 field L for the running example with K a quadratic extension giving 336. If the closing slide is to show field size as the difficulty parameter, that column has to be produced rather than read off. Worth knowing before the slide is drafted, since it is a computation, not a lookup.
[C] And the parameter may be the right one for a reason the deck can state. X153 sits at 300 digits with a degree-336 field; X110 needs about 10800 digits with a degree-288 field. So the field is SMALLER and the precision is enormously larger, which rules out field degree alone as the explanation and makes the pair of numbers more interesting than either. Your sentence gives the actual mechanism: the polynomial of degree 288 is deducible by Galois theory but cannot be finished off, so the cost is in the recovery step, not in the size of the field it lands in.
[C] That mechanism ties the closing slide back to slide 13 and to minima-uv3.246. Slide 13 is the isomorphism problem: build the embedding through compatible complex embeddings and a Vandermonde solve, with the height-reduction trick taking 120k digits down to 4k. X110 is that step failing, at a degree where the reduction does not save you. And the Elsenhans-Steel deflation work is aimed at precisely this class of polynomial, which is why the "triggered a new algorithm" oral line and the stuck example are the same story rather than two.
[C] One discrepancy to resolve before any count is printed. astra reports the corpus as "exactly 161 S4-orbits", while INDEX.md has 189 rows. Those cannot both be the number of surfaces on a slide. Likely the census carries rows beyond the S4-orbit representatives, or was built before a deduplication, but I have not established which. Do not print a corpus size until that is settled; the safe phrasing is your own prospective one, without a count.
[C] IMPORTANT, and it bears on minima-uv3.243 rather than on this comment. astra contradicts using the paper's abstract as a completion claim on slide 19: the latest census still has M2 absent, six new rows lack Picard and Aut metadata, and X156 is unfinished. So the abstract's "We determine, rigorously, the Picard lattices of all smooth five-nomial quartics" is the paper's goal, not today's status, and the slide must use your own prospective phrasing, "we are trying to cover all the K3 surfaces given by five nomials". That is what you dictated, so the two comments agree; the point is that the abstract cannot be quoted as the stronger version.
>>[minima-uv3.243] AUTHOR (2026-09-15): slide 19: we should make clear that we are trying to cover all the K3 surfaces given by five nomials
[C] The scope sentence already exists, in your own abstract. paper/fivenomials.tex:24, verbatim: "We determine, rigorously, the Picard lattices of all smooth five-nomial quartics in projective three-space with all non-zero coefficients equal to $1$. We also compute the Galois action on the Picard lattices and compute the algebraic part of the Brauer groups." The commented-out title at :19 is "Picard lattices of five-nomial quartic K3 surfaces". Nothing in the deck says any of this; the word "five-nomial" appears nowhere in Lecture 3 except in the directory name of the source.
[C] It also retroactively explains the running example, which the deck currently presents as a one-off. x^4 + xyzw + y^3z + yw^3 + z^3w is five monomials with every coefficient 1, so it is a member of the corpus, not a specimen chosen for being interesting. Slide 9 introduces it as "a running example inspired by Klein-Mukai" and the room has no way to know it is one entry in a census. Saying the scope at the end reframes everything between 9 and 18 as a worked instance of a programme.
[C] And it closes the course's own arc, which is the strongest reason to say it here rather than anywhere else. The corpus subsection at :78 records that the rank-1 list has 8 examples up to symmetry, for instance x^3w + xz^3 + y^4 + yw^3 + z^4, and that "$\rho = 1$ on each is established by [EC2021] (Costa-Sertöz crystalline obstruction)". That is Lecture 2's method, the obstruction map of its slide 9, doing the low-rank end of the same census that Lecture 3's periods do at the high-rank end. Three lectures, one corpus, said in one sentence on the last slide.
[C] What is still open in the corpus, if you want the ask to be sharper than "do you have one". The same subsection says completeness of the rank-1 list needs \rho > 1 on every other fivenomial, and concedes a real limitation: "our algorithm can return an empty sublattice on those examples without small-degree rational curves". The gap is currently filled by Wim Nijgh's argument, that either \Aut(X) is non-trivial, hence \rho > 1 by Huybrechts Cor. 15.2.12 for quartics, or one of the four coordinate-plane intersections already contains a line or a conic. Thomas Bouchet classified the fivenomials. So the honest ask is for a surface the method does not reach, which is one with no small-degree rational curves, rather than for any hard surface.
[C] PROVISIONAL, flagged rather than used: that subsection is an \emre{TODO ...} block, so the "8 examples up to symmetry" figure and the Nijgh argument are notes toward a subsection, not settled text. The abstract at :24 is a finished sentence and is the safe thing to quote. If a number goes on the slide it should be one you have checked today.
[C] Consequence for what this slide is. It is currently a bare <section class="center"> with an h2 and no body at all, and its notes say "Leave this slide up during questions" and that it "should be the thing left on the screen during questions". Now that minima-uv3.242 drops slide 20, this really is the last thing the room looks at. Recommend one line above the question and nothing else, for instance "We are doing this for every smooth five-nomial quartic in P^3 with unit coefficients." Two sentences and it stops being a closing slide and becomes a summary slide, which slide 18 already is.
[C] Alternative worth considering, since you may not want text on the closer at all: put the scope line at the top of slide 9 instead, where the example is introduced, and leave 19 as the bare question. That gets the framing in earlier, where it changes how the room reads the next ten slides, rather than at the end where it only recontextualises. Your call; it is a question of whether the scope is the setup or the sign-off.
What it does. You said the lectures exist to get people working on these problems. The deck already ends by asking the room for work. It should be its own slide rather than a trailing line, and it should be the thing left on the screen during questions.
20 PROPOSED DROP: What is open, in one place
new
Collect what the three lectures left behind, since a listener who wants to start cannot be expected to have kept a list:
- Does End(Ā) = Z force Ā_p ∼ E² infinitely often? (1.4)
- Is there a way to get a sharp upper bound on the Picard number of a K3 surface? The double cover with RM by Q(√2) is the known obstacle, the case where the method has not yet succeeded. (2.2)
- How does one compute a good enough approximation to Frob_p to obstruct the last two cycles? (2.2)
- Can Gal(K/Q) be computed, and is it Aut Λ? (3.3)
- What is H¹(Gal, Pic X̄), and what does it give for Brauer-Manin? Over a number field it is Br₁(X)/Br₀(X), which is where the algebraic Brauer-Manin obstruction comes from. (3.3)
- Is there a practical saturation algorithm for lattices generated by low-degree rational curves? (3.3)
Proposed, not taken. This slide is mine and it is the one place I have added something the decks do not contain. It follows from the purpose you named, but it is an addition to your course and it needs your yes.
What it does. Collects the unanswered questions from all three lectures in one place.
PROPOSED DROP: this slide is unapproved, repeats questions already left open in their lectures, and comes after the Leiden deck's own last line. Removing it gives 19 spoken slides and leaves "Do you have a challenge K3 surface for us?" on screen during questions.
>>[minima-uv3.242] AUTHOR (2026-09-15): Drop Slide 20, we will not present these.
[C] This answers minima-uv3.2, "L3 s20: approve the collected open-problems slide", which has been blocked on you since 2026-09-13 with the note "Speaker notes say 'Proposed, not taken' and 'needs your yes'." The answer is no. It also settles the first item under "Open on this lecture". The decision is recorded on the bead; the bead stays open until the commit that actually removes the slide, per the review rule in artifacts/orch/RULES.md.
[C] Reading applied: drop the collection, keep the questions that live elsewhere. "These" is the six collected problems on this one slide. Slides 17 and 19 also end on open questions, and you approved both, slide 17's purpose being in your own framing "three questions handed to a room full of people who could answer them", slide 19 being the request for a challenge surface. So this is not an instruction to stop presenting open problems, and I have not read it as one.
[C] Nothing is lost except the collection, which was the drop rationale already on the page. Of the six, three are slide 17's and 18's, two are Lecture 2's slides 19 and 20, and one is Lecture 1's. Each is stated where it arises.
[C] EXCEPT one clause, and it is the consequence flagged under minima-uv3.230 now coming true. Slide 20 is the only place in the course that says what H^1(Gal, Pic X-bar) is FOR. Its wording: "What is $H^1(\operatorname{Gal}, \operatorname{Pic}\overline{X})$? Over a number field it is $\operatorname{Br}_1(X)/\operatorname{Br}_0(X)$, and from there the algebraic Brauer-Manin obstruction." Slide 17 asks the same question bare, as $H^1(\operatorname{Gal}(\bar k/k), \operatorname{Pic} \overline{X}) = ?$, and I checked: neither "Brauer" nor "Br_1" appears anywhere on slide 17. Meanwhile this plan's slide 8 entry says "The Brauer-Manin payoff is deferred to slide 20". With 20 gone, that deferral points at nothing: Lecture 1's slide 5 states the isomorphism and promises it matters for rational points, and Lecture 3 never cashes it. Recommend moving that one clause onto slide 17, which costs a line and closes the loop. If slide 8 also sheds its Brauer displays under minima-uv3.230, this becomes the only surviving statement of it in Lecture 3.
[C] Good consequence, and it was your own stated intent. Slide 19's entry says the challenge-surface line "should be its own slide rather than a trailing line, and it should be the thing left on the screen during questions". Slide 20 sat after it and took that away. Dropping 20 restores the course's last line to the Leiden deck's own last line.
[C] Count, cumulative across this pass, and it no longer reaches your target. Starting from 20 content slides: minus this one is 19; splitting slide 9 under minima-uv3.231 makes 20; splitting slide 10 under minima-uv3.232 makes 21; the title slide under minima-uv3.227 makes 22 positions in the built deck, 21 of them spoken. artifacts/plan/plan.md line 3 records your own budget, "50 + 45 + 45 minutes; at about 25 slides per hour that budgets roughly 21 + 19 + 19 slides", so 21 spoken against 19 is about five minutes over. Dropping 20 was the single reduction available on the page and it is now spent. Any further reduction has to come from a merge, and the two obvious candidates are the pairs that already share a title: slides 11 and 12, both "Reconstructing quadric surfaces", and slides 16 and 17, both "Computing the Galois action".
[C] Moot now: minima-uv3.13 recorded that "L3 slide 20: the (1.4), (2.2), (3.3) markers are 0.7em at 0.7 opacity". That observation dies with the slide.