The General Geometry of 6/9: An Expanded Atlas of K3 Surfaces

1. The finite classification result

Up to the dihedral symmetries of the square, there are exactly 16 patterns of type 6/9. For each one, six roots satisfy three independent quadrics. The order below is determined by the local complexity of the preferred basis of conditions:

\mathrm{red}<\mathrm{yellow}<\mathrm{blue}.

This complexity order is the reverse of the editorial color preference used in the general atlas, where red is preferred to yellow and yellow to blue. Within the RRY profile, the parallel case precedes the intersecting one: after parametrizing the two red conics, it is completely solved by one equality of tf.

Partition of the sixteen types

16=3_{\mathrm{RRR}}+2_{\mathrm{RRY,parallel}} +6_{\mathrm{RRY,intersecting}} +2_{\mathrm{RYY}}+2_{\mathrm{RYB}}+1_{\mathrm{YYB}}.

2. Expanded atlas: from simple to difficult

Three red progressions

The simplest local conditions: every quadric is a rational progression conic.

ABDEFH →RRR

three progressions, two shared centers

Reduction: a correctly twisted quartic
Surface: 4I₄+4I₂
Geometric MW rank: 1≤r≤2
Sections: 1 native
Coverage: proved rational chart; global coverage undetermined

ABDEFJ →RRR

a chain of three progressions

Reduction: a palindromic quartic
Surface: 4I₄+4I₂
Geometric MW rank: 1≤r≤2
Sections: 1 native
Coverage: proved rational chart; global coverage undetermined

ABEFGH →RRR

a triangle of pairwise means

Reduction: a pullback of the Legendre family
Surface: 4I₄+4I₂
Geometric MW rank: 1≤r≤2
Sections: 1 native
Coverage: proved rational chart; global coverage undetermined

Two parallel red conics and a yellow gluing

A particularly simple global case: the third quadric becomes equality of tf exactly.

ABEFGJ →RRY

parallel progressions containing E

Reduction: complete tfmn normalization
Surface: Km(E×E)
Geometric MW rank: —
Sections: complete tfmn description
Coverage: all nondegenerate rational points

ABDFHJ →RRY

parallel progressions omitting E

Reduction: linearly the same tfmn space
Surface: Km(E×E)
Geometric MW rank: —
Sections: complete tfmn description
Coverage: all nondegenerate rational points

III

Two intersecting red conics and a yellow gluing

After parametrizing the two progressions, a pointed genus-one quartic remains.

ABCDEH →RRY

two progressions sharing H

Reduction: a residual quartic
Surface: 2I₄+8I₂
Geometric MW rank: 2≤r≤4
Sections: 2 independent
Coverage: proved rational chart; global coverage undetermined

ABCDEJ →RRY

two progressions sharing J

Reduction: a residual quartic
Surface: 2I₄+8I₂
Geometric MW rank: 1≤r≤4
Sections: 1 native
Coverage: proved rational chart; global coverage undetermined

ABCDFH →RRY

two progressions sharing H

Reduction: a palindromic quartic
Surface: 2I₄+8I₂
Geometric MW rank: 1≤r≤4
Sections: 1 native
Coverage: proved rational chart; global coverage undetermined

ABCDHJ →RRY

two progressions sharing D

Reduction: a palindromic quartic
Surface: 2I₄+8I₂
Geometric MW rank: 1≤r≤4
Sections: 1 native
Coverage: proved rational chart; global coverage undetermined

ABCEGH →RRY

shared center E; first reading

Reduction: a quartic shared with ABCEGJ
Surface: 2I₄+8I₂
Geometric MW rank: 2≤r≤4
Sections: 2 independent
Coverage: proved rational chart; global coverage undetermined

ABCEGJ →RRY

shared center E; second reading

Reduction: a quartic shared with ABCEGH
Surface: 2I₄+8I₂
Geometric MW rank: 2≤r≤4
Sections: 2 independent
Coverage: proved rational chart; global coverage undetermined

One red conic and two yellow norms

One rational progression links two independent Gaussian factorizations.

ABCDEG →RYY

the central CEG progression

Reduction: an even quartic
Surface: 12I₂
Geometric MW rank: 1≤r≤6
Sections: 1 native
Coverage: proved rational chart; global coverage undetermined

ABCDGJ →RYY

the BDJ progression and two norms

Reduction: an even quartic
Surface: 12I₂
Geometric MW rank: 1≤r≤6
Sections: 1 native
Coverage: proved rational chart; global coverage undetermined

Red, yellow, and blue norms

A Gaussian norm and an x²+2y² norm must both agree with a progression.

ABCDEF →RYB

three conditions on the CDE block

Reduction: an even quartic
Surface: 12I₂
Geometric MW rank: 1≤r≤6
Sections: 1 native
Coverage: proved rational chart; global coverage undetermined

ABCDFG →RYB

three norms without the center

Reduction: an even quartic
Surface: 12I₂
Geometric MW rank: 1≤r≤6
Sections: 1 native
Coverage: proved rational chart; global coverage undetermined

Two yellow and one blue norm without red

The locally hardest type: no quadric is parametrized in advance by a progression.

ABCGHJ →YYB

the unique red-free type

Reduction: a smooth (2,2,2) intersection; Gaussian chart
Surface: 4I₄+4I₂
Geometric MW rank: 1≤r≤2
Sections: 1 native
Coverage: proved rational chart; global coverage undetermined

For all sixteen types, infinite rational constructions and exact positive specializations of type exactly 6/9 are published. In the table, r means rank E(Qbar(t)) for the chosen Jacobian fibration, not the rank of every specialized fiber. No such rank is assigned to the two Kummer patterns: their proved global result is the complete tfmn description of all nondegenerate rational solutions.

\operatorname{Km}(E\times E),\qquad E:\ v^2=u^3-u.

A shared passport records the same types and multiplicities of singular fibers in the chosen fibration. It does not by itself prove that the corresponding K3 surfaces are isomorphic, birational, or members of one deformation family with the stated additional structure.

3. Three passports and the rank ladder

The fourteen nonparallel patterns fall into three semistable elliptic K3 passports:

Passport	Patterns	Root lattice	Upper bound
12I₂	4	A₁¹²	r≤6
2I₄+8I₂	6	A₃²⊕A₁⁸	r≤4
4I₄+4I₂	4	A₃⁴⊕A₁⁴	r≤2

For the relatively minimal elliptic models with a section and no multiple fibers, the Euler numbers in every row sum to 24 and the canonical-bundle formula gives χ(O)=2. Once smoothness of the minimal model is checked, these are K3 surfaces. The Shioda–Tate formula and ρ(K3)≤20 give the following ladder of geometric-rank upper bounds:

\begin{aligned} 12I_2:\;&20-2-12=6,\\ 2I_4+8I_2:\;&20-2-(2\cdot3+8)=4,\\ 4I_4+4I_2:\;&20-2-(4\cdot3+4)=2. \end{aligned}

The more of the Picard lattice is occupied by components of singular fibers, the less room remains for the free group of sections.

4. In what sense K3 models arise

Six selected square entries give six roots and three homogeneous quadrics. If their projective intersection in P5 is smooth, adjunction gives a trivial canonical bundle and H1(O)=0 follows from Lefschetz or the Koszul complex; the surface is then K3. In the other articles, K3 refers to the smooth minimal model of the Jacobian of a chosen genus-one fibration or, for the parallel pair, to the minimal resolution of a Kummer surface.

These constructions explain the systematic appearance of K3 geometry, but they do not identify the original intersection of quadrics, an individual chart, a Jacobian surface, and a Kummer model. The articles use explicitly stated birational maps and minimal resolutions between the relevant models.

5. The congruent family really has no generic non-torsion section

Regard k as an independent parameter and consider

\mathcal C_k:\qquad y^2=x^3-k^2x.

Its invariants are

\Delta=64k^6,\qquad c_4=48k^2.

The minimal elliptic surface over ℙ¹ₖ has two I₀* fibers, at k=0 and k=∞. Their root lattice D₄⊕D₄ has rank 8 and their Euler numbers sum to 12. This is a rational elliptic surface, hence

\operatorname{rank}\mathcal C_k(\overline{\mathbb Q}(k)) =10-2-8=0.

At k=1 the fiber is y²=x³−x, whose rational torsion group is (Z/2Z)^2. Specialization of generic torsion into this good fiber is injective, while the three nonzero 2-torsion points are already visible over Q(k). Thus rank zero together with specialization determines the entire group of generic sections.

All generic sections over ℚ(k)

\mathcal C_k(\mathbb Q(k)) =\{\mathcal O,(0,0),(k,0),(-k,0)\} \cong(\mathbb Z/2\mathbb Z)^2.

Thus trivial sections do exist—the zero section and full 2-torsion—but there is no generic non-torsion section.

6. The function-field extension in F4+

F4+ does not find sections over the original k-line. It replaces ℚ(k) by the function field of the two-dimensional surface

\mathcal S:\qquad x^2-1=\rho^2y(x^2-y^2).

On this surface, the common area parameter A=x(x²−1) already has two different representations. Hence the same congruent-number curve C_A acquires two independent points:

\begin{aligned} R_1&=\bigl(x^2(x^2-1),\,x^2(x^2-1)^2\bigr),\\ R_2&=\bigl(\rho^2x^2(x^2-y^2),\, \rho^3x^2(x^2-y^2)^2\bigr). \end{aligned}

\operatorname{rank}C_A(\mathbb Q(\mathcal S))\ge2.

Separately, the normalized F4+ cubic in the parameter tau=rho^2 forms a rational elliptic surface of exact arithmetic and geometric rank 2. The quadratic base change tau=rho^2 gives an elliptic K3 surface: its trivial lattice and the two lifted independent sections prove that its geometric rank is also exactly 2, with no new free sections over Qbar(rho). The universal curve C_A over Q(S) is a different elliptic object.

For ABEFGJ and ABDFHJ, the proved tfmn normalization identifies the nondegenerate rational part of the 6/9 problem with the corresponding model S. Thus F4+ coordinates give a global description for these two patterns and serve as a comparison construction for the others.

7. Do the other surfaces have F9+ analogues?

Two meanings must be distinguished.

Meaning	Definition	Status
Weak	An explicit rational curve or section on the K3 surface generating a one-parameter 6/9 family.	Available for all 16 types
Strong	A special base curve C→ℙ¹ after which the fibration rank strictly increases.	Proved for the F9+→F4+ branch; not assessed for the other 14 models

In the original F9+ branch, the curve s²=x³−2x gives the substitution ρ=1/s and adds a third independent section to the two F4+ sections. Hence the generic rank after this base change is at least 3.

The Jacobians of the other fourteen patterns already have explicit non-torsion sections over Q(t), without an additional base change. These sections are not strong F9+ analogues: they belong to the original Mordell–Weil group and do not by themselves prove a rank jump after base change.

8. A program for finding strong analogues

For each fibration E_t, a strong F9+ analogue should consist of a curve C and a map t=t(C) for which a section appears

\left[P_{\mathrm{new}}\right]\notin E(\mathbb Q(t))\otimes_{\mathbb Z}\mathbb Q \subset E(\mathbb Q(C))\otimes_{\mathbb Z}\mathbb Q.

The most natural search order is the reverse of the atlas color order:

Start with the 4I₄+4I₂ surfaces with 1≤r≤2: one new independent section would immediately prove exact rank 2.
Then treat 2I₄+8I₂: search for second and third sections through low-degree quadratic base changes.
Finally treat 12I₂: the upper bound 6 leaves more possible directions and requires a broader classification of multisections.

Candidates should arise from tangent parabolas to the quartics, low-degree quadratic substitutions, and conditions forcing one of the three remaining entries to become a square. The last option ties the search for sections directly to the transition 6/9→7/9.

9. What this says about the full 9/9 problem

On every 6/9 surface, the three omitted entries are rational functions D₁,D₂,D₃. A full square requires the simultaneous lift

u_1^2=D_1,\qquad u_2^2=D_2,\qquad u_3^2=D_3.

The 9/9 problem is thereby reformulated exactly as the search for rational points on a simultaneous triple quadratic cover of one of the 6/9 surfaces, together with distinctness and positivity conditions. Infinite 6/9 families do not determine where a possible obstruction to 9/9 lies; it may be global, local, or absent.

The completeness of the sixteen-type classification, the three passports, the rank ladder, native sections, and the tfmn equivalence of the parallel pair are proved. Exact ranks for most K3 surfaces and strong F9+ analogues for the fourteen nonparallel types are not yet asserted.

Open the card atlas of 6/9 →F4+: the universal surface F9+: rank-raising substitutions Previous chapter: ABCGHJ