Type-(1,3) double covers, Picard-twelve K3 surfaces and Gaussian Hodge conjectures

1. Contribution, dependencies and scope

Let $(B,L)$ be a complex abelian surface with a symmetric polarization of type $(1,3)$. Normalize the symmetric linearization so that $h^0(L)^+=2$ and $h^0(L)^-=1$. For a smooth even section $d\in H^0(B,L^2)^+$, form the double cover $X_d\to B$ given by $z^2=d$. Write $\delta$ for its sheet involution, and choose the other inversion lift $\tau$, so that the selected space

$$ U_X=H^2(X_d,\mathbb Q)^{\tau=1,\delta=-1} $$

has Hodge numbers $(1,8,1)$.

Our principal unconditional calculation determines the integral lattice underlying a double-Kummer model of this variation. Its full anti-invariant lattice and the primitive complement of its six persistent exceptional differences are

$$ P\simeq U_{\mathrm{hyp}}^{\oplus2}\oplus D_{12}(-1), \qquad R\simeq U_{\mathrm{hyp}}^{\oplus2}\oplus D_6(-1). \tag{1.1} $$

Here $U_{\mathrm{hyp}}$ is the integral hyperbolic plane; $D_n(-1)$ is the negative root lattice. Both lattices have determinant $4$. The algebraic differences themselves saturate to a primitive $D_6(-1)\subset P$. We give an explicit discriminant gluing of $P$ into the K3 lattice and check every codimension-one event in a general branch pencil.

The comparison conclusion uses precisely stated external inputs. OpenAI's September 30, 2026 preprint [1], Lemma 5.5 and Theorem 6.1, assert, respectively, realization of integral K3 period data over the original curve and an algebraic comparison by two square substitutions. Assuming these two statements, our geometric calculations yield the following result.

Conditional comparison theorem. For a sufficiently general fixed $(B,L)$ and a sufficiently general pencil in $\mathbb P H^0(B,L^2)^+$, there is a smooth projective K3 family $S\to O$ over a nonempty open of the original pencil. At all but countably many points $b$ of a nonempty open, an algebraic correspondence

$$ \Gamma_b\in\operatorname{CH}^2(S_b\times X_b)_{\mathbb Q} $$

induces a Hodge similarity

$$ T(S_b)_{\mathbb Q}\longrightarrow U_{X_b}, \qquad (\Gamma_bx,\Gamma_by)_{X_b}=2(x,y)_{S_b}. \tag{1.2} $$

At very general points the integral transcendental and Néron–Severi lattices of this realization are

$$ T(S_b)\simeq R,\qquad \operatorname{NS}(S_b)\simeq U_{\mathrm{hyp}}\oplus D_{10}(-1). \tag{1.3} $$

An additional countability argument produces algebraic comparisons with the same rational normalization over a nonempty Zariski open of the full ten-dimensional parameter space. This last existence statement does not specify a simultaneous integral marking.

These generic comparison conclusions do not construct a curve or abelian source for $U_X$. The rank-eight quadratic-locus theorem of [1] does not apply to this rank-ten space. Neither the general rational Hodge conjecture nor algebraicity of a Kuga–Satake tensor on this larger locus follows.

Separately, §9 proves an unconditional source statement on a four-dimensional locus of the K3 target: its transcendental cohomology embeds by an algebraic correspondence into the tensor product of a fixed Gaussian CM elliptic curve and a Prym fivefold. This uses an explicit classical isotrivial source, Hermitian-form classification and algebraicity of rational K3 Hodge isometries. It does not depend on the two preprint inputs. It does not produce a source for the generic eight-dimensional K3 locus. Section 10 supplies the remaining Gaussian tenfold by an uniruled quotient; Appendix B gives an explicit line curve, incidence projector and comparison on a nonempty open. Section 10 supplies the mixed determinant classes by a specialization of Schoen's cycles. This proves ordinary HC on all mixed powers of the Gaussian K3, elliptic curve, Prym fivefold and constructed tenfold, including the specified even-Clifford Kuga–Satake tensors. Generalized HC follows at full-group points from fundamental sources and at special points from the complete spin restriction. A direct Prym period calculation and odd-rank Hermitian similitude transport the abelian statements to any Gaussian fivefold of signature $(1,4)$, including endomorphism and Picard jumps. The same K3 isogeny gives a Chow motive of abelian type on this locus. Within the open set reached by the conditional comparison, the source also transfers to the selected transcendental part of an original $X$ whenever $U_X$ admits the indicated orthogonal Gaussian action.

The enlargement method itself is not new. Proposition 3.1 of [1] already retains a rank-sixteen anti-invariant lattice for its rank-eight seed; its Lemma 5.2 gives a general equivariant completion, and its local analysis supplies the two roots at an unbranched Kummer node. We extend that framework to the type-$(1,3)$ family and determine the exact discriminant, primitive saturation and target. The K3 moduli space in (1.3), including a concrete double-quadric presentation, is already described by Ma [2, §9.5 and Appendix A.1].

The rank-ten seed and its rational form already appear in lelong_ridge [3, Remark 5.2], together with numerical evidence for period rank eight. That paper proves a different rank-eight seed result and explains the failure of the indicated Gauss half-sum construction under its stated monodromy assumption. Our Appendix A replaces the numerical period evidence with a complete argument; §§3–4 determine the integral lattices and primitive saturation, §9 supplies the Gaussian source and motivic conclusion, and §10 gives the actual missing spin source, determinant transport and mixed support. Appendix B constructs the explicit Gaussian line cylinder and identifies it with the specified quotient invariant. We use none of the conditional rank-eight Hodge-conjecture claims of [3] as an input.

The finite computations supplied here check the Kummer code, discriminant gluing, local matrices and one explicit target model. They do not certify the external comparison theorems. The written period-rank argument in Appendix A is included because it is needed to choose pencils without a hidden simplicity assumption.

2. The actual double-Kummer cover

The normalized linearization of $L$ has positive sign at ten points $J_{10}\subset B[2]$ and negative sign at the other six points $J_6$. This follows from the nondegenerate Arf-zero quadratic refinement on $B[2]\simeq\mathbb F_2^4$, equivalently from the holomorphic Lefschetz trace $h^0(L)^+-h^0(L)^-=1$. The lift $\tau$ is the negative of this linearization on the cover coordinate. It fixes the two points above each element of $J_6$ and exchanges the sheets above $J_{10}$.

Let $K=\operatorname{Km}(B)$ be the resolved Kummer surface, with exceptional curves $N_a$, $a\in B[2]$. Resolve the twelve nodes of $X/\tau$ and blow up its ten smooth points lying over $J_{10}$. The result $\widehat Y$ has a finite double cover

$$ p:\widehat Y\longrightarrow K $$

branched on

$$ C+\sum_{a\in J_{10}}N_a=2A,\qquad C^2=12,\quad A^2=-2. \tag{2.1} $$

For a general $d$, these eleven branch components are smooth and disjoint; $g(C)=7$. At $a\in J_6$, the curve $N_a$ lifts to two disjoint $(-2)$-curves $E_{a,+},E_{a,-}$, exchanged by $\delta$. At $a\in J_{10}$, the ramification curve $R_a$ has square $-1$ and $2R_a=p^*N_a$. These assertions follow in local coordinates from the two possible inversion lifts on $z^2=d(x,y)$; they also explain why the ten extra blowups add only sheet-invariant cohomology.

The quotient construction gives $q(\widehat Y)=0$: on $X$, $H^1(X,\mathbb Q)=H^1(B,\mathbb Q)$ by ample vanishing, and inversion acts as $-1$. The two invariant holomorphic two-forms consist of the form from $B$ and the selected odd theta form. Thus $p_g(\widehat Y)=2$. Equation (2.1) gives

$$ e(\widehat Y)=2\cdot24-\bigl(-12+10\cdot2\bigr)=40. $$

Consequently $b_2(\widehat Y)=38$ and the free integral second cohomology has signature $(5,33)$.

Write

$$ M=H^2(\widehat Y,\mathbb Z)^\delta/\mathrm{torsion}, \qquad P=H^2(\widehat Y,\mathbb Z)^{-\delta}/\mathrm{torsion}, $$

where these mean eigensublattices in free cohomology. Pullback identifies $M_{\mathbb Q}$ with $H^2(K,\mathbb Q)(2)$, so $M$ has rank $22$ and signature $(3,19)$; $P$ has rank $16$ and signature $(2,14)$. Put $\beta_a=[E_{a,+}]-[E_{a,-}]$ for $a\in J_6$. They are mutually orthogonal with square $-4$, and

$$ P_{\mathbb Q}=U_Y\perp\bigoplus_{a\in J_6}\mathbb Q\beta_a. \tag{2.2} $$

The resolved quotient graph induces a flat Hodge isomorphism $\beta:U_Y\to U_X$. The degree-two projection formula gives

$$ (\beta x,\beta y)_X=2(x,y)_{\widehat Y}. \tag{2.3} $$

It is this particular similarity, rather than an unspecified rational isometry of abstract quadratic spaces, that is later made algebraic.

3. Exact discriminant and equivariant completion

3.1. Saturation of the invariant lattice

Let $L_0=p^*H^2(K,\mathbb Z)\simeq\Lambda_{\mathrm{K3}}(2)$, and let $M_0$ be generated by $L_0$ and the ten ramification classes $R_a$. Then

$$ [M_0:L_0]=2^{10}. \tag{3.1} $$

Indeed, a relation among these ten classes modulo $L_0$ would make the half-sum of a nonempty subset of the corresponding $N_a$ integral on $K$. The primitive Kummer-lattice theorem says that its possible nonzero supports, modulo the root lattice, are the thirty affine hyperplanes of $\mathbb F_2^4$ and the full set of sixteen points [4, Proposition 2.2 and Remark 2.3]. No affine hyperplane lies in the Arf-zero set $J_{10}$: its intersection with that set has size $4$ or $6$. Thus no relation exists.

It follows that $|\det M_0|=2^{22}/2^{20}=4$. The only possible strict integral saturation $M_0\subset M$ would have index $2$ and make $M$ unimodular. But $P$ is even: $K_{\widehat Y}=p^*A$ is invariant, so $xK_{\widehat Y}=0$ for $x\in P$, and Wu's parity formula gives $x^2\equiv0\pmod2$. Primitive orthogonal complements in the unimodular free cohomology have discriminant groups of equal order. A unimodular $M$ would therefore make $P$ even unimodular of signature $(2,14)$, contrary to the signature congruence modulo $8$. Hence

$$ M=M_0,\qquad |\det P|=4. \tag{3.2} $$

3.2. An elementary identification of $P$

The bilinear discriminant group of $L_0$ is $H^2(K,\mathbb Z)/2H^2(K,\mathbb Z)$. Adjoining the ten $R_a$ takes a bilinear-isotropic subgroup $W$ of dimension $10$, and $A_M=W^\perp/W\simeq(\mathbb Z/2)^2$. Every vector of $M^*$ lies in $L_0^*=\frac12p^*H^2(K,\mathbb Z)$ and has integral square because $H^2(K,\mathbb Z)$ is even. Thus the discriminant bilinear form of $M$ has zero diagonal in $\mathbb Q/\mathbb Z$. Unimodular gluing gives the same property for $P$.

Since $P$ is even, its discriminant quadratic form $q_P$ takes integral values modulo $2\mathbb Z$. Its nondegenerate alternating bilinear form on $(\mathbb Z/2)^2$ ensures that some order-two class $c$ has $q_P(c)=1$. The subgroup generated by $c$ is isotropic for the bilinear form, so adjoining it gives an integral index-two overlattice $P'$. This lattice is odd and unimodular. The classification of indefinite odd unimodular lattices gives $P'\simeq I_{2,14}$ [5, Chapter 14, Corollary 1.3(ii)]. Because $P$ is even and of index two, it is exactly the even part of $P'$. Ma's explicit computation of that even part [2, Example A.1] gives

$$ P\simeq U_{\mathrm{hyp}}^{\oplus2}\oplus D_{12}(-1). \tag{3.3} $$

In particular its discriminant form is the anisotropic form on $(\mathbb Z/2)^2$: all three nonzero values are $1$ modulo $2\mathbb Z$. This proof avoids a general classification of indefinite 2-elementary lattices.

3.3. Completion preserving the original pencil

Take $F=U_{\mathrm{hyp}}\oplus D_4(-1)$. It has signature $(1,5)$ and the same anisotropic discriminant form, which also equals its negative. Gluing $P$ and $F$ along a discriminant anti-isometry produces an even unimodular lattice of signature $(3,19)$, hence $\Lambda_{\mathrm{K3}}$. Both factors embed primitively. If $e,f$ is the standard basis of the added hyperbolic plane, then $h=e+f$ is primitive and has square $2$.

For fixed $B$, the lattice $M$ is a constant local system: it is generated by pullbacks from the fixed Kummer surface and the ten labeled ramification curves. Its discriminant action is therefore trivial. Equivariance of unimodular gluing shows that monodromy of $P$ is trivial on $A_P$ too. It extends to $\Lambda_{\mathrm{K3}}$ by the identity on $F$, fixing $h$. This construction leaves no finite cover of the original pencil in the monodromy data.

4. The primitive $D_6$ and the rank-ten complement

For every pair $a,b\in J_6$ there is an affine hyperplane whose intersection with $J_6$ is precisely $\{a,b\}$. Its other six points lie in $J_{10}$. Pull back the associated integral Kummer half-sum and subtract the six ramification classes. The result proves that $p^*(N_a+N_b)/2$ is integral. Hence

$$ \frac{\beta_a+\beta_b}{2} =E_{a,+}+E_{b,+}-\frac{p^*(N_a+N_b)}2 \tag{4.1} $$

is integral and anti-invariant. Subtracting $\beta_b$ gives the half-difference too.

Identify $\beta_a$ with twice a negative unit coordinate vector. The vectors in (4.1) and their half-differences generate $A=D_6(-1)$. Its nonzero discriminant values are $1,\frac12,\frac12$ modulo $2\mathbb Z$. Thus it has no proper even overlattice and is primitive in $P$.

Put $R=A^\perp\subset P$ and $k=[P:A\perp R]$. Since $|\det P|=|\det A|=4$, the determinant relation gives $|\det R|=k^2$. Primitive gluing makes $k$ divide $4$. Moreover $\beta_a/2\in P^*$: for $x\in P$,

$$ (\beta_a,x)=2(E_{a,+},x). $$

Its class in $A^*/A$ is the vector class $v$, with $q_A(v)=1$. The gluing subgroup projected to $A^*/A$ therefore lies in $v^\perp=\{0,v\}$. Thus $k\leq2$. The case $k=1$ would make $R$ even unimodular of signature $(2,8)$, which is impossible. Consequently

$$ k=2,\qquad |\det R|=4. \tag{4.2} $$

The nonzero glue class has projections $v$ and $c\in A_R$, with $c$ of order two and $q_R(c)=-q_A(v)=1$ modulo $2\mathbb Z$. Adjoin $c$ to $R$. As in §3.2, this gives an odd unimodular lattice $R'\simeq I_{2,8}$ whose even part is $R$. Therefore

$$ R\simeq U_{\mathrm{hyp}}^{\oplus2}\oplus D_6(-1), \qquad R_{\mathbb Q}\simeq U_{\mathrm{hyp},\mathbb Q}^{\oplus2} \perp\langle-1\rangle^{\oplus6}. \tag{4.3} $$

The full family has period rank eight by Appendix A. At a very general point $R_{\mathbb Q}=U_Y$ has no rational $(1,1)$-class; therefore $A$ is the entire algebraic saturation inside $P$. The primitive embedding $R\subset\Lambda_{\mathrm{K3}}$ has complement of signature $(1,11)$ with discriminant form $-q_R$. That form again has an order-two class of value $1$. Its odd unimodular extension is $I_{1,11}$, so its even part is $U_{\mathrm{hyp}}\oplus D_{10}(-1)$. This proves the claimed integral target in the explicit pencil realization.

These are abstract lattice isometries. We do not claim that six vanishing vectors transported from different singular fibers are pairwise orthogonal: their intersections depend on the transport paths, and the local root calculation alone does not determine them.

5. Boundary of a general pencil and its Hodge line

The invariant system $H^0(B,2L)^+$ has dimension eight. Its morphism realizes the singular Kummer surface as a genus-seven model in $\mathbb P^7$, with precisely the sixteen Kummer nodes contracted [4, §5.3]. A general hyperplane pencil has only ordinary tangencies in the smooth locus and transverse passages through individual nodes. The codimension-two locus of multiple contacts, higher contacts, nontransverse node passages and simultaneous events is avoided. This is the usual characteristic-zero Lefschetz-pencil assertion, applied on the smooth locus and separately to the sixteen node hyperplanes. A leading member avoiding these loci is chosen at infinity.

There are three finite local models.

An ordinary contact away from the Kummer nodes. The local double surface is $z^2=xy+t$. Its vanishing cycle is an integral $(-2)$-root in $P$, and monodromy is its Picard–Lefschetz reflection.

A passage through $a\in J_6$. Upstairs, equivariant Morse coordinates give

$$ z^2=x^2+y^2+t,\qquad \tau(x,y,z)=(-x,-y,z),\qquad \delta(z)=-z. \tag{5.1} $$

On a Kummer-resolution chart $a=x^2$, $v=y/x$, the double surface becomes $z^2=a(1+v^2)+t$. At $t=0$ the branch is $a=0$ together with a residual curve meeting it at $v=\pm i$. There are two disjoint ordinary double points, giving two orthogonal integral anti-invariant roots $r_1,r_2$ in the nearby smooth $\widehat Y$. Local monodromy is $s_{r_1}s_{r_2}$.

Equivalently the unresolved quotient has the $A_3$ model $uv=(w^2-t)^2$. In a negative $A_3$ basis $\alpha_1,\alpha_2,\alpha_3$, with adjacent pairing $1$, one may take

$$ r_1=\alpha_1+\alpha_2,\qquad r_2=\alpha_2+\alpha_3. $$

The involution and monodromy on this local lattice are

$$ \alpha_1\longleftrightarrow\alpha_3,\qquad \alpha_2\longmapsto-\alpha_1-\alpha_2-\alpha_3. \tag{5.2} $$

The persistent difference $r_1-r_2$ and transverse vector $r_1+r_2$ each have square $-4$. Keeping only their transverse projection hides the integral roots needed for the K3 Weyl condition.

To fix the signs geometrically, $\tau$ in (5.1) preserves the orientation of the real vanishing sphere and $\delta$ reverses it; $\delta$ also exchanges the two persistent exceptional curves. The action of $w\mapsto-w$ on the minimal resolution of the central surface $uv=w^4$ is not a substitute for this nearby calculation: that action need not lift to the chosen simultaneous resolution over the parameter, and the Weyl correction changes the marking.

A passage through $a\in J_{10}$. Here $\tau(x,y,z)=(-x,-y,-z)$ in the upstairs quadratic model. It acts antipodally on the vanishing sphere, so no vanishing class survives in the $\tau$-invariant cohomology. The quotient smoothing at the minimal surface is the rational-homology-ball smoothing of $\frac14(1,1)$. Its added ramification curve is sheet-invariant. For example, the Kummer chart has equation

$$ \eta^2=a\bigl(a(1+v^2)+t\bigr),\qquad \eta=xz. $$

Thus the monodromy on $U_Y$ is the identity; the six exceptional difference lines are unaffected as well. The action on $P$ is the identity. We do not describe this event as an ADE degeneration of the double-Kummer model.

These local calculations determine the entire monodromy operator: outside the disjoint node neighborhoods the original family is topologically trivial, and (2.2) accounts for all additional anti-invariant resolution classes. The nontrivial finite operators are therefore the stated products of integral root reflections.

There are $28$ ordinary contacts, six $J_6$ passages and ten $J_{10}$ passages. For the first count, the pencil on $K$ has twelve base points, and the blown-up total surface has Euler number $36$. The smooth genus-seven fibers contribute $2(-12)=-24$. At each of the sixteen node passages, the fiber contains a rational bridge meeting the residual curve in two points; its Euler defect is $2$. The remaining defect is

$$ 36-(-24)-16\cdot2=28. \tag{5.3} $$

The count is a consistency check as well as a complete boundary list.

At infinity the monodromy on $P$ is $\delta=-1$; it extends as the identity on $F$. Its square is the identity on the completed K3 lattice. At finite singular parameters the roots are orthogonal to $h$ and to the limiting period.

For completeness, interior extension of the period is checked upstairs. After the quadratic local cover $t=v^2$, an $A_1$ smoothing can be written $xy=(z-v)(z+v)$. Blowing up $(x,z-v)$ gives charts

$$ z=v+xa,\quad y=a(2v+xa), \qquad z=by-v,\quad x=b(by-2v), $$

which are smooth over $v$. The relative two-form $dx\wedge dz/x$ becomes $dx\wedge da$ in the first chart and $db\wedge dy$ in the second. Thus the selected theta two-form extends crepantly. It remains a nonzero global form on the central resolved surface because it is nonzero away from the node neighborhoods. Its polarized norm stays positive. The same upstairs invariant limiting class treats the noncanonical quotient event. The exceptional difference classes remain algebraic, so (2.2) transfers this interior period to $P$ and to its K3 completion.

Now make the square substitution $t=s+u^2$, with $s$ away from the finite boundary. The upstairs compactification over $\mathbb P^1_u$ uses the line bundle $L\boxtimes\mathcal O(1)$ and branch section

$$ (d_0+sd_1)u_0^2+d_1u_1^2. $$

The selected anti-sheet relative canonical direct image is $H^0(B,L)^-\otimes\mathcal O(1)$, a line bundle $\mathcal O(1)$. Explicitly, for $v=1/u$ and $w=z/u$,

$$ w^2=d_1+(d_0+sd_1)v^2,\qquad \frac{\theta\,dx\wedge dy}{z} =v\frac{\theta\,dx\wedge dy}{w}. \tag{5.4} $$

There are no additional zeros or poles at finite parameters by the crepant calculation. Finite monodromies fix the nonzero limiting form, so invariant-frame descent adds no fractional character. Thus the prescribed extended Hodge line of the completed period data is exactly $\mathcal O_{\mathbb P^1}(1)$.

6. Application of the two external comparison inputs

We specify the imported assertions to separate them from our geometric work. Lemma 5.5 of [1] takes an integral K3 local system, a primitive positive flat class, and a positive isotropic holomorphic period line with finite local monodromy and interior limits. It realizes these data by polarized ADE K3 surfaces over the original curve. When the local monodromies are in the Weyl groups of the limiting perpendicular roots, its extended model has relative dualizing sheaf pulled back from the Hodge line. Its final assertion supplies a smooth K3 family on an open of the original curve, preserving a rational Hodge summand perpendicular to every persistent root. The period-stack contraction input is related to the published construction in Di Lorenzo–Fringuelli–Vistoli [6, Lemma 1.12]; [6] does not itself assert our comparison.

Appendix A and the ordered choice of a very general abelian surface followed by a sufficiently general line allow the pencil to avoid containment in every proper rational Hodge locus. Consequently $U_Y$ is simple and has no rational $(1,1)$-class except at countably many parameters. Every locally flat root that remains algebraic on an open has zero projection to $U_Y$: its projection would otherwise be a persistent rational $(1,1)$-class. Analytic continuation and a very general point force that projection to vanish.

The hypotheses of Lemma 5.5 are now supplied by §§3–5. We obtain a smooth K3 family $S\to O$ on the original pencil, with an isometric copy of $U_Y$. Its generic integral transcendental lattice is $R$, and its generic Picard lattice is (1.3).

For every admissible $s$, square substitution kills the monodromy at infinity. At the remaining punctures the monodromies are products of roots perpendicular to the limiting period and $h$. The same lemma gives an ADE K3 model over $\mathbb P^1_u$ with

$$ K_Z=f^*\mathcal O_{\mathbb P^1}(-1), \tag{6.1} $$

by (5.4). On a smooth projective resolution, intersection of the canonical class with the square of an ample pullback is negative: exceptional divisors have zero such intersection because their images have codimension at least two. Thus the canonical class is not pseudo-effective, and the uniruledness criterion of Boucksom–Demailly–Păun–Peternell [7] applies.

It is necessary that this is the square pullback of the fixed $S$-family, rather than a different family with equal periods. Use the finite Galois cover chosen in the first application of Lemma 5.5 and normalize its fiber product with $t=s+u^2$. On this common cover the extended period maps and their equivariance data are literal pullbacks. Functoriality of the period-stack contraction and fully faithful GAGA identify the algebraic ADE families and their group actions. Effective Galois descent then identifies their generic fibers. The generic resolution of the original contraction remains birational after base change. Hence the uniruledness in (6.1) belongs to a model of the actual square-pulled-back $S$-family. This is the common-cover argument of [1, §5.5], applied to our completed local system.

Theorem 6.1 of [1] compares such a K3 family with a surface family carrying commuting involutions whose projector annihilates cohomology outside degree two. It requires a specified rational Hodge isomorphism, nonconstant simple K3-type variation, and uniruled threefold models after the square substitutions. It allows a similarity rather than an isometry.

For $X\to O$ take the literal algebraic projector

$$ e_X=\frac14(1+\tau)(1-\delta). \tag{6.2} $$

The involution $\delta$ fixes $H^0,H^4$ and $H^1(X)=H^1(B)$; it also fixes $H^3$ by duality. Thus (6.2) annihilates every degree except two and has image $U_X$ there. After $t=s+u^2$, the total space $z^2=d_0+(s+u^2)d_1$ is birational to a conic bundle over $B$. A smooth general conic over $\mathbb C$ is rational, and these fibers cover the total space; a smooth projective model is therefore uniruled.

All hypotheses of Theorem 6.1 are satisfied with the specified map $\beta$ in (2.3). It gives a codimension-two correspondence inducing exactly $\beta_b$ at all but countably many points of a nonempty open. Equations (1.2)–(1.3) follow. This is an application of the two named preprint results, not an independent proof of their global descent or cylinder arguments.

7. A full-base existence statement by countable spread

We explain the quantifier improvement separately, because a pencil chosen through a prescribed parameter would not by itself control the exceptional countable set produced afterward.

Take an irreducible fine level parameter space $\mathcal B$ for the smooth type-$(1,3)$ family. It has dimension $3+7=10$. The cover and its involutions give the relative projector $e_X$. Enumerate polarized K3 surfaces by their countably many Hilbert schemes. Enumerate rational codimension-two cycles on their products with $X$ using relative Hilbert schemes, finite products and rational coefficient lists. Smooth connected stratifications give countably many finite-type strata, each carrying an actual K3 family and an actual cycle $\Gamma$.

On each stratum impose the equalities of flat cohomological maps

$$ e_X\Gamma=\Gamma,\qquad \Gamma\Gamma^t=2e_X \quad\text{on }H^2(X,\mathbb Q). \tag{7.1} $$

Flatness means that a connected stratum satisfies (7.1) everywhere or nowhere. Call the former strata successful. Their images in $\mathcal B$ are constructible.

The very general fixed-pencil comparisons give successful points. To see this with no extra algebraic component of the cycle, project the source onto its transcendental cohomology using the usual algebraic projectors for $H^0,H^4$ and the Néron–Severi lattice, and project the target by $e_X$. The resulting map is the similarity $\beta$ on the whole transcendental part and zero on its orthogonal complement. Equation (2.3) gives $\beta\beta^t=2e_X$, establishing (7.1).

Suppose all successful images were nondense. Their closures, together with the proper Hodge loci needed for simplicity, would be countably many proper algebraic subsets of $\mathcal B$. First choose very general $(B,L)$ whose branch fiber is not contained in any of them. Such choices exist: a proper closed subset cannot contain the generic full branch fiber of the irreducible total parameter space. Then choose a line in that fiber contained in none of these subsets and satisfying §5. Each proper closure meets this line in finitely many points. Their union covers at most countably many points of the line. But §6 supplies successful comparisons at all but countably many points of an open in that line, an uncountable set. This is a contradiction.

Thus one successful stratum dominates $\mathcal B$. Its constructible image contains a nonempty Zariski open $\mathcal B^0$. At every point of $\mathcal B^0$, (7.1) supplies an algebraic embedding of $U_X$ into K3 cohomology with the corresponding normalization. Its transpose has multiplier $2$:

$$ (\Gamma^t x,\Gamma^t y)_S=2(x,y)_X \quad (x,y\in U_X). $$

The image contains the holomorphic two-form. Its Hodge orthogonal complement has only type $(1,1)$, so the K3 transcendental part is exactly the nonalgebraic part of this image. At a very general parameter it has rank ten and the K3 has Picard number twelve.

No simultaneous K3 family on all of $\mathcal B^0$ was assumed. A finite generically étale cover supporting one can be obtained after this final spread, by taking a closed point of the generic fiber of a successful stratum and shrinking the base. This final cover is different from leaving an unspecified cover in the original-pencil argument, where uniruledness could be lost. The spread preserves rational cohomological data; it does not by itself retain the exact integral lattice marking of §4.

8. The known target and a reproducible geometric control

Ma identifies the locus in (1.3) as the two-elementary K3 moduli space $\mathcal M_{12,2,1}$ [2, §9.5, Appendix A.1]. Let $Q=\mathbb P^1\times\mathbb P^1$. Take a smooth $(3,3)$ curve $C$ with a point $p$ at which one ruling fiber $F_1$ is triply tangent and the other ruling fiber $F_2$ is transverse. The resolved double cover branched on $C+F_1+F_2$ is a K3; the branch has a $D_8$ singularity at $p$ and two further nodes. Ma proves that this presentation is birational to the eight-dimensional moduli space. Its genus-four fixed curve has a totally ramified trigonal point.

For a concrete check, write

$$ \begin{aligned} C(x,y)&=y^3+xP_1(y)+x^2P_2(y)+x^3P_3(y),\\ P_1(y)&=1+2y+3y^2+5y^3,\\ P_2(y)&=7+11y+13y^2+17y^3,\\ P_3(y)&=19+23y+29y^2+31y^3, \end{aligned} $$

and take $w^2=xyC(x,y)$. The attached checker verifies smoothness of the projective $(3,3)$ curve on all four quadric charts modulo $1009$, which implies smoothness in characteristic zero. It checks the total tangency and the two additional simple intersections. Binary-quartic invariants for the $x$-ruling give

$$ II^*+2I_2+10I_1, $$

with smooth infinity and visible lattice $U_{\mathrm{hyp}}\oplus E_8(-1)\oplus A_1(-1)^{\oplus2}$. The $j$-invariant is nonconstant.

The second ruling makes the $D_{10}$ lattice visible directly. The substitution, suggested by gabrielov_cell,

$$ \widehat X=y^4/x,\qquad \widehat W=y^4w/x^2 $$

gives the exact equation

$$ \widehat W^2=\widehat X^3+yP_1(y)\widehat X^2+ y^5P_2(y)\widehat X+y^9P_3(y). \tag{8.1} $$

The checker verifies (8.1), orders $(2,3,12)$ of $c_4,c_6,\Delta$ at $y=0$, and a squarefree residual discriminant polynomial of degree twelve. Thus this ruling has $I_6^*+12I_1$ and trivial lattice $U_{\mathrm{hyp}}\oplus D_{10}(-1)$.

For this explicit sample the computations prove only $\rho\geq12$; they do not establish a Picard-rank upper bound or Mordell–Weil rank zero. For a very general member those conclusions follow from Ma's moduli theorem, and the lack of a proper even overlattice of the trivial lattice excludes torsion. The presentation and the moduli theorem are known results, not our new K3 family. No correspondence from this model to an abelian variety is obtained from these elliptic-fiber calculations alone.

9. An algebraic source on the Gaussian locus

The source family used here is classical: its elliptic model is Artebani–Sarti's Example 8.2(i) [8]. The point of the following argument is to transport that source to the specified rational quadratic space and then to an order-four locus in Ma's model.

Proposition 9.1. Let $Z$ be a projective K3 surface. Suppose $V\subset H^2(Z,\mathbb Q)$ is a nondegenerate rational Hodge subspace containing $H^{2,0}(Z)$, with quadratic form

$$ V\simeq R_{\mathbb Q} =U_{\mathrm{hyp},\mathbb Q}^{\oplus2}\perp\langle-1\rangle^{\oplus6}. $$

Suppose $V$ has an orthogonal rational Hodge endomorphism $J$ with $J^2=-1$. Then an algebraic correspondence embeds $T(Z)_{\mathbb Q}$ into

$$ H^1(E_i,\mathbb Q)\otimes H^1(P,\mathbb Q), \tag{9.1} $$

where $E_i$ has complex multiplication by $\mathbb Q(i)$ and $P$ is a five-dimensional Prym of a genus-seven cyclic-four curve over a genus-two curve. In particular (9.1) embeds all of $V$ when $V=T(Z)_{\mathbb Q}$. No assertion about embedding every additional Tate class of $V$ into this particular tensor is needed.

9.1. The Hermitian form, with its local invariants

Let $q$ be the cup-product form on $V$, and let $K=\mathbb Q(i)$ act through $J$. Replace $J$ by $-J$ if necessary so that it acts by $i$ on $H^{2,0}$, using the same convention for the source below. Then

$$ h(x,y)=q(x,y)+i\,q(x,Jy) $$

is a $K$-Hermitian form, linear in the first argument, of signature $(1,4)$. Its real part is $q$. Diagonalize $h=\langle a_1,\ldots,a_5\rangle$ over $K$, with $a_j\in\mathbb Q^\times$. The underlying rational form is the orthogonal sum of $\langle a_j,a_j\rangle$. Its Hasse invariant at every rational place is therefore

$$ \epsilon(q)=\prod_{j=1}^5(a_j,a_j) =\left(\prod_{j=1}^5a_j,-1\right). \tag{9.2} $$

Here $(\ ,\ )$ denotes the Hilbert symbol; cross terms between two repeated pairs occur four times and cancel.

But $R_{\mathbb Q}\simeq I_{2,8}$, whose Hasse invariant is $(-1,-1)^{28}=1$. Thus the determinant of $h$ is a norm from $\mathbb Q(i)$ at every place and hence globally. Equivalently, the associated quaternion algebra splits, or the conic $x^2+y^2=d z^2$ has a rational point for $d=\prod a_j$. Landherr's classification [9], also stated in [10, Theorem 70], now gives

$$ h\simeq\langle1,-1,-1,-1,-1\rangle. \tag{9.3} $$

Thus all the orthogonal $K$-structures in this proposition have the same Hermitian form. Matching the rational quadratic discriminants alone would not have justified this step.

9.2. A classical product source with an open period image

Choose seven distinct points $a,b_1,\ldots,b_6$ of $\mathbb P^1$, and put

$$ f(t)=(t-a)^2\prod_{j=1}^6(t-b_j),\qquad G:\ v^4=f(t). $$

The normalization of $G$ has genus seven: the six simple branch points have ramification defect three and the double point has defect two, so $2g(G)-2=-8+6\cdot3+2=12$. The quotient by $v\mapsto-v$ is the genus-two curve

$$ D:\ s^2=\prod_{j=1}^6(t-b_j),\qquad s=v^2/(t-a). $$

Its Prym $P=\operatorname{Prym}(G/D)$ has dimension five. The holomorphic eigenspace dimensions for the three nontrivial fourth-root characters are $1,2,4$, up to reversing the primitive characters. Thus the primitive Gaussian part has signature $(1,4)$.

Take $E_i:z^2=u^3+u$. There is an actual rational map of degree four

$$ E_i\times G\dashrightarrow S_f,\qquad (u,z;t,v)\longmapsto(v^2u,v^3z,t), \tag{9.4} $$

where

$$ S_f:\quad Y^2=X^3+f(t)X. $$

The diagonal order-four action $(u,z,v)\mapsto(-u,iz,iv)$ fixes the coordinates in (9.4). The minimal resolution is an elliptic K3 with fibers $6III+I_0^*$ [8, Example 8.2(i)] and order-four automorphism $(X,Y,t)\mapsto(-X,iY,t)$.

The visible trivial lattice has rank twelve and rational form

$$ U_{\mathrm{hyp},\mathbb Q}\perp \langle-2\rangle^{\oplus6}\perp D_4(-1)_{\mathbb Q} \simeq U_{\mathrm{hyp},\mathbb Q}\perp\langle-1\rangle^{\oplus10}. \tag{9.5} $$

For the last equality, $D_4(-1)_{\mathbb Q}\simeq\langle-1\rangle^4$ and each pair $\langle-2,-2\rangle$ is rationally isometric to $\langle-1,-1\rangle$. Fiber inversion, the square of the order-four automorphism, fixes this trivial lattice pointwise: it fixes the components of each $III$ and $I_0^*$ fiber and the zero section. Its orthogonal complement is anti-invariant. More precisely, modulo the fiber, section and nonidentity fiber components, the elliptic Leray decomposition identifies $H^2$ with $H^1(\mathbb P^1,j_*R^1\pi_*\mathbb Q)$, where $j$ includes the smooth-fiber locus. Inversion is $-1$ on $R^1\pi_*$ and therefore on this quotient. Since it fixes the nondegenerate trivial lattice pointwise, its orthogonal complement is the negative eigenspace, including at Picard jumps. Thus the rank-ten complement carries an orthogonal $\mathbb Q(i)$-structure.

The moduli of seven points with the double point distinguished has dimension four. An elliptic isomorphism recovers this configuration from its singular fibers, up to an automorphism of $\mathbb P^1$. The map to elliptic K3 moduli is therefore generically finite. Forgetting the fibration still has zero-dimensional fibers: a fixed K3 has countably many Néron–Severi classes and hence countably many choices of fiber and section classes, and each fiber class determines its pencil. A finite-type fiber with countably many complex points has dimension zero. After choosing a local polarization or level chart these are ordinary algebraic moduli maps.

The unmarked K3 image consequently has dimension four. K3 local Torelli and the Gaussian period-domain bound show that the generic transcendental rank is exactly ten and the period image is open in its four-dimensional complex ball. Witt cancellation of (9.5) inside $\Lambda_{\mathrm{K3},\mathbb Q} =U_{\mathrm{hyp},\mathbb Q}^{\oplus3}\perp\langle-1\rangle^{16}$ gives $R_{\mathbb Q}$ for the rank-ten complement. Its form stays constant on a local parameter chart, even at Picard jumps. By §9.1 its Hermitian form is (9.3).

9.3. Exact period transport and algebraic correspondences

The rational points of the group $U(h)$ are dense in $U(1,4)(\mathbb R)$. To see this without a density assumption on arithmetic subgroups, use the rational Cayley map

$$ A\longmapsto (1-A)(1+A)^{-1} $$

on the rational vector space of skew-Hermitian endomorphisms. It parametrizes a neighborhood of the identity. Rational points in that vector space are real-dense; the closure of $U(h)(\mathbb Q)$ therefore contains a neighborhood of the identity. The real group $U(1,4)$ is connected, so the closure is the whole group.

Choose one simply connected source chart with a fixed marking whose period image contains a nonempty ball-open set $\mathcal O$. Given the period $p$ of $V$, transitivity of the real unitary action and rational density give $g\in U(h)(\mathbb Q)$ with $g(p)\in\mathcal O$. There is an actual source $S_f$ with this period. This produces an exact rational Hodge isometry from $V$ to the rank-ten complement $V_f$ in $H^2(S_f,\mathbb Q)$. Density is used to enter an open set, not to infer an algebraic cycle by taking a limit. The resulting correspondence is existential: this argument does not give an algorithm recovering the seven branch points from an arbitrary input K3 surface.

Witt's extension theorem extends this isometry to the full rational K3 cohomology. Both complementary spaces have only type $(1,1)$, since $V$ and $V_f$ contain the holomorphic two-forms, so the extension is a Hodge isometry. Buskin's theorem [11, Theorem 1.1] makes it an algebraic correspondence.

Resolve (9.4) as $f:\widetilde{E_i\times G}\to S_f$ with blowdown $p:\widetilde{E_i\times G}\to E_i\times G$. Pullback is injective because $f_*f^*=4\operatorname{id}$ on $H^2(S_f,\mathbb Q)$. The exceptional summands of the blowup are Tate, so projection by $p_*$ loses no transcendental class. The transcendental part maps into the $H^1(E_i)\otimes H^1(G)$ Künneth summand: a morphism from the transcendental K3 Hodge structure to any Tate summand is zero. This follows from polarizability and the absence of rational $(1,1)$-classes in $T(S_f)$. Moreover the involution $v\mapsto-v$ on $G$ lifts fiber inversion on $S_f$, so the image lies in $H^1(E_i)\otimes H^1(G)^-$. The algebraic projector $(1-\tau^2)/2$ on $G$ has image $H^1(P)$. The usual Poincaré and isogeny correspondences identify this with the Prym factor algebraically. This proves (9.1), including at Picard jumps by restriction to the transcendental summand.

9.4. The stronger motivic consequence

Corollary 9.2. Under the hypotheses of Proposition 9.1, the rational Chow motive of $Z$ is of abelian type and is finite-dimensional in Kimura's sense.

Here ``of abelian type'' means belonging to the pseudoabelian tensor subcategory of rational Chow motives generated by motives of curves. The full rational Hodge isometry constructed in §9.3 is an isogeny of K3 surfaces. Huybrechts's motivic isogeny theorem [12, Theorem 0.2] lifts it to an isomorphism $h(Z)\simeq h(S_f)$ of rational Chow motives. For the resolved product map, the graph identity $\Gamma_f\circ{}^t\Gamma_f=4\Delta_{S_f}$ holds in the Chow group. Thus $h(S_f)$ is a direct summand of $h(\widetilde{E_i\times G})$. The latter is a sum of the product motive $h(E_i)\otimes h(G)$ and Tate motives from point blowups. It is of abelian type and Kimura finite-dimensional; these properties pass to direct summands [13]. This proves the corollary. The use of [12] is essential here: an arbitrary cohomological injection alone would not justify the Chow-motive conclusion.

9.5. A four-dimensional locus in Ma's family

In the quadric model impose

$$ C=y^3+x(a_0+a_2y^2)+x^2(b_1y+b_3y^3)+x^3(c_0+c_2y^2). \tag{9.6} $$

Then $\sigma(x,y,w)=(-x,-y,iw)$ satisfies $\sigma^2=\delta$ and acts by $-i$ on $dx\wedge dy/w$. The smooth locus of these curves is nonempty: the sample $(a_0,a_2,b_1,b_3,c_0,c_2)=(1,2,3,5,7,11)$ is smooth on all four quadric charts modulo $1009$. It also has a unique totally ramified point for the $x$-trigonal map. Writing the cubic in $y$ as $a y^3+b y^2+c y+d$, the common triple-root conditions $3ac-b^2=27a^2d-b^3=0$ have greatest common divisor $x$ in this sample; the leading-coefficient and infinity cases give no additional triple root. Thus the hypotheses of Ma's open-immersion presentation [2, Proposition A.5] hold on a nonempty open.

There are seven coefficients before projectivization, including the coefficient of $y^3$. The continuous quadric automorphisms preserving the selected point, rulings and simultaneous-sign involution are the two coordinate scalings. Their quotient has dimension four. Forgetting the selected involution has finite fibers, because a smooth genus-four curve has finite automorphism group. Ma's presentation therefore gives a four-dimensional K3 moduli image. The anti-invariant lattice is the same rank-ten $R$ as before, and $\sigma$ squares to $-1$ on it. Its ball has dimension four, so the generic transcendental rank remains ten and the generic Picard number is twelve. Proposition 9.1 supplies the algebraic source for its transcendental cohomology throughout this smooth open, also when the Picard rank increases. Corollary 9.2 supplies the corresponding statement for its Chow motive.

The scope should be distinguished from the following already known or unproved assertions. Ordinary HC for powers of a K3 whose transcendental endomorphism field is CM is already known [14, Corollary 2.3]. The isotrivial family and the classification of order-four automorphisms are classical [8]. Hermitian classification in geometric source constructions also has precedents in Paranjape [15] and Ingalls–Logan–Patashnick [16], in different orthogonal and unitary ranks. The half-twist formalism and the product-source method are classical; van Geemen [17, Theorem 3.10 and Example 3.11] treats the related Eisenstein isotrivial family. An especially close Gaussian precedent is Nori's domination of cyclic quartic K3 surfaces by an elliptic curve times a curve, recorded in van Geemen [18, Example 11.3]. There a tangent-line construction gives an isotrivial family of Gaussian elliptic curves, which becomes a product after finite base change. Our result instead specifies transport for the rank-ten rational form and Ma sublocus, with the particular genus-seven cyclic source and its Prym fivefold. It is not a new general elliptic-product domination or half-twist principle. We do not identify a prescribed full even-Clifford Kuga–Satake tensor from (9.1), and we make no generalized-HC claim from that embedding alone. The additional geometric constructions of Section10 supply ordinary HC and the specified KS tensors and generalized HC, including at smooth Gaussian special points. Finally, the Gaussian sublocus has dimension four; it does not supply a source at a generic point of the eight-dimensional orthogonal period domain.

9.6. Conditional transfer to the original selected cohomology

Let $\mathcal B^0$ be the nonempty open furnished by §7, under the two comparison assumptions. Suppose $X=X_b$, $b\in\mathcal B^0$, and $U_X$ admits an orthogonal rational Hodge endomorphism $J$ with $J^2=-1$. Choose the actual K3 surface and correspondence of (7.1), and put $V=\operatorname{im}(\Gamma^t|_{U_X})$. This is a nondegenerate rank-ten Hodge subspace containing the K3 holomorphic two-form. The quadratic normalization is

$$ q_V\simeq 2q_{U_X}\simeq4q_R\simeq q_R. $$

The second equality uses the actual quotient similarity (2.3); the last is the rational rescaling by $1/2$. The Gaussian action transports to $V$ by $\Gamma^t$, with inverse $\Gamma/2$ on its image. Proposition 9.1 therefore applies to this particular comparison K3, without requiring an integral marking or identifying it with a member of Ma's displayed chart.

Write $U_X^{\mathrm{tr}}$ for the orthogonal complement in $U_X$ of its rational $(1,1)$-classes. Its image under $\Gamma^t$ is precisely the K3 transcendental part. Composition with the correspondence in Proposition 9.1 yields an algebraic embedding

$$ U_X^{\mathrm{tr}}\hookrightarrow H^1(E_i,\mathbb Q)\otimes H^1(P,\mathbb Q). $$

The selected projector and the divisor projectors are algebraic, so the statement concerns actual correspondences on $X$ as well. This conclusion is conditional on §7 and applies within $\mathcal B^0$; it asserts no comparison at an excluded special fiber and no source for the generic orthogonal variation.

These conditional applications are nonvacuous. On a marked local chart where the original period map has rank eight, fix one orthogonal $J_0^2=-1$ on $R_{\mathbb Q}$, using its model $I_{2,8}$. The positive $J_0$-complex lines form a four-ball inside the orthogonal period domain. The rational orthogonal group is dense in the identity component of its real group, by the same Cayley argument as in §9.3, and that real component acts transitively on the connected orthogonal period domain. Consequently the union of rational translates of this four-ball is analytically dense. Every translate has an orthogonal rational Gaussian action. Pulling back by the submersive period map gives an analytically dense countable union of local six-dimensional loci to which the conditional source statement applies. Density supplies no correspondence at a generic orthogonal period.

10. Actual Gaussian spin sources and mixed Hodge conjectures

This section is independent of the two comparison inputs in §6. We use $E=E_i$ and the cyclic curve $G$ of §9.2, with primitive Prym $P$. Write $W=H^1(P,\mathbb Q)$. Choose its complex Gaussian character so $W_\sigma$ has Hodge numbers $(1,4)$, and let $e=H^{1,0}(E)$. Thus the selected K3 space is

$$ V_\mathbb C=(e\otimes W_\sigma)\oplus (\bar e\otimes W_{\bar\sigma}). \tag{10.1} $$

The involution, character and alternating projectors used below are rational algebraic correspondences. A displayed complex eigenspace always denotes one constituent of its indicated rational conjugate pair.

Theorem 10.1 (Gaussian mixed support). There exists an abelian variety $A$ of dimension ten and an algebraic correspondence inducing

$$ H^1(A,\mathbb Q)(-1)\simeq \left(e\otimes\bigwedge^2 W_\sigma\right) \oplus\left(\bar e\otimes\bigwedge^2 W_{\bar\sigma}\right) \subset H^3(E\times P\times P,\mathbb Q). \tag{10.2} $$

The mixed determinant space

$$ \left(\bigwedge^5 W_\sigma\otimes e^{\otimes3}\right) \oplus\left(\bigwedge^5 W_{\bar\sigma}\otimes\bar e^{\otimes3}\right) \subset H^8(P\times E^3,\mathbb C) \tag{10.3} $$

is the complexification of a rational space of algebraic cycle classes. Both statements hold for every smooth branch configuration of §9.2.

Put $C=E\times P\times A$. The ordinary rational Hodge conjecture holds for every $S_f^r\times C^n$, at every smooth branch configuration. The same holds after adjoining powers of the actual Kuga–Satake variety, and every specified rational tensor defining its usual Kuga–Satake correspondence is algebraic. These statements transfer to the K3 surfaces of Proposition 9.1 through their rational Hodge-isometric source, including at Picard jumps.

The generalized rational Hodge conjecture holds for all these mixed products as well, including at Picard and endomorphism jumps. Here generalized HC means that every rational Hodge substructure of $H^k$ of level at most $k-2p$ is supported in codimension at least $p$. We first prove the full-group case in §§10.5–10.7; Section10.8 extends ordinary HC and the prescribed KS class, and Section10.9 calculates actual support for every smaller Gaussian Hodge group.

Corollary 10.2 (arbitrary Gaussian fivefolds). Every complex abelian fivefold $B$ with a polarization-compatible Gaussian action of signature $(1,4)$ is Gaussian-linearly isogenous to a Prym $P$ above. Thus (10.2) and (10.3) transport to $B$. Ordinary and generalized HC hold for all powers of $E\times B\times A$ and for the associated Gaussian K3 mixed products, without a full-group hypothesis.

The abstract spin decomposition and half twists are due to van Geemen [17, Proposition 3.8 and Theorem 3.10]. Our additional geometric steps are a symmetric-square quotient giving actual support for its tenfold, and a compact-type specialization of Schoen's existing determinant cycles [18, Theorem 2.0 and Corollary 3.1]. Related hypersurface half-twist constructions appear in van Geemen–Izadi [19, §§4–5]. The fully twisted source strategy belongs to the work of Abdulali [20, Theorem 10]; we supply the connecting cycles in this case. His later survey [21, Appendix B.5] corrects the 2004 type-IV theorems by requiring $|p-q|=1$. Its Appendix A(4) lists fivefold signature $(3,2)$. Consequently those results do not by themselves imply the present signature-$(1,4)$ assertion. HC for powers of the K3 alone is already covered by [14]; the added content concerns the mixed abelian sources, generalized support and the prescribed Kuga–Satake tensors.

10.1. A direct open Prym chart and isogeny transport

The primitive holomorphic forms on $G$ have bases

$$ \omega=\frac{dt}{v},\qquad \eta_j=\frac{t^j(t-a)\,dt}{v^3},\quad 0\le j\le3, \tag{10.4} $$

for the $-i$ and $i$ characters. To check regularity, at a simple branch point use $t-b=s^4$, $v=s$ times a unit; at a double branch point use $t-a=s^2$, $v=s$ times a unit. At infinity $v=t^2$ times a unit, which gives exactly the bound $j\le3$. The remaining two holomorphic forms have character $-1$ and belong to the genus-two quotient.

For equivariant deformations the cotangent space of the seven-point configuration modulo $\operatorname{PGL}_2$ is $H^0(\mathbb P^1,K_{\mathbb P^1}^{\otimes2}(D_7))$, of dimension four. The dual differential of the curve period map is multiplication of holomorphic forms; restricting to the primitive Gaussian factors gives

$$ \omega\eta_j= \frac{t^jdt^2}{(t-a)\prod_{j'=1}^6(t-b_{j'})}. \tag{10.5} $$

These four sections are a basis. Descent of invariant quadratic differentials changes the pairing by a nonzero trace scalar only. Thus the Prym period map has rank four and supplies an actual open chart in the unitary ball, without an appeal to the Picard number of an individual K3 surface.

Let $h_0$ be the rational Hermitian form on this marked chart, and $h$ that of $H^1(B)$ in Corollary 10.2. Both have signature $(1,4)$. A compatible polarization exists by averaging any polarization under the order-four action. Their determinants $d_0,d$ in rational Gaussian bases are positive. Set $c=d_0/d>0$. Then

$$ \frac{\det(ch)}{\det h_0}=c^4=N_{\mathbb Q(i)/\mathbb Q}(c^2). \tag{10.6} $$

Scaling a polarization by $c$ preserves positivity. Landherr's classification [9] and [10, Theorem 70] identifies $ch$ with $h_0$. The rational unitary density argument of §9.3 sends the target period into this actual open chart. It gives an exact Gaussian-linear rational weight-one Hodge isomorphism, hence an algebraic isogeny of abelian varieties after clearing denominators. Isogeny graphs and rational inverses transport all the correspondences below. The use of odd Hermitian rank in (10.6) is essential; this is not a determinant normalization available in general even rank.

10.2. The symmetric-square quotient is uniruled

Let $M=E\times G\times G$, and retain the automorphisms $\epsilon(u,z)=(-u,iz)$ and $\tau(t,v)=(t,iv)$. The group

$$ H=\langle(\epsilon,\tau,1),(\epsilon,1,\tau),s_{12}\rangle \simeq(\mathbb Z/4)^2\rtimes S_2 \tag{10.7} $$

has order $32$. Before the interchange, its quotient has function field

$$ Y^2=X^3+f(t_1)f(t_2)X,qquad X=(v_1v_2)^2u,quad Y=(v_1v_2)^3z. \tag{10.8} $$

Adjoining $v_1,v_2$ recovers the product field and has degree sixteen. The interchange fixes $X,Y$ and exchanges $t_1,t_2$. In coordinates $[Z:S:P]$ on $\operatorname{Sym}^2\mathbb P^1=\mathbb P^2$, its full fixed field is

$$ \mathcal Y:\quad Y^2=X^3+F_8X,qquad F_8=\ell_a^2\prod_{j=1}^6\ell_{b_j},quad \ell_c=P-cS+c^2Z. \tag{10.9} $$

The lines are tangent to the discriminant conic $S^2-4PZ=0$.

The standard projective Weierstrass model has fundamental line bundle $L=\mathcal O_{\mathbb P^2}(2)$; adjunction gives

$$ \omega_{\mathcal Y}=\pi^*(\omega_{\mathbb P^2}\otimes L) =\pi^*\mathcal O_{\mathbb P^2}(-1). \tag{10.10} $$

It is integral and normal. Indeed, on an affine base chart its singular equations imply $Y=0$, $3X^2+F_8=0$ and $X\,dF_8=0$. Together with its equation these force $X=F_8=0$, so the singular locus has codimension at least two. The infinity section is smooth. As a hypersurface it is Cohen–Macaulay, and Serre's criterion applies.

Let $r:Y\to\mathcal Y$ be a projective resolution and let $D$ be an ample Cartier divisor on $\mathcal Y$. Every exceptional divisor maps into codimension at least two, hence intersects $r^*D^2$ trivially. Since $\pi^*\mathcal O(1)$ is a nonzero effective divisor,

$$ K_Y\cdot r^*D^2=K_{\mathcal Y}\cdot D^2<0. \tag{10.11} $$

A pseudoeffective divisor has nonnegative intersection with the square of a nef divisor. Thus $K_Y$ is not pseudoeffective, and [7, Corollary 0.3] implies that $Y$ is uniruled. No crepant resolution is assumed.

10.3. From invariant cohomology to an actual tenfold

The two fourth-root actions in (10.7) impose the character match

$$ H^3(M,\mathbb C)^H= (e\otimes\bigwedge^2W_\sigma) \oplus(\bar e\otimes\bigwedge^2W_{\bar\sigma}). \tag{10.12} $$

Geometric interchange of two degree-one factors acts by minus tensor interchange; this is why the exterior square occurs. The other Künneth summands in degree three have no invariants. Call the rational space in (10.12) $\mathcal W$. It has dimension twenty, Hodge numbers $h^{2,1}=h^{1,2}=10$, and Gaussian character dimensions $(4,6)$ after Tate twist by one. Its projector is the algebraic group average.

For a smooth projective uniruled threefold $Y$, $H^3(Y)=N^1H^3(Y)$. To see the actual support, take a generically finite rational domination $Q\times\mathbb P^1\dashrightarrow Y$ by a smooth projective surface $Q$ and resolve it. The terms $H^3(Q)$ and $H^1(Q)(-1)$ are supported on divisors: for the first, use hard Lefschetz and the Gysin map from a hyperplane curve. Resolving a rational map between smooth threefolds by smooth-center blowups adds only curve $H^1(-1)$ terms to third cohomology. Pushforward is surjective because its composite with pullback is the generic degree. This proves the support assertion by correspondences.

We must transfer that support through a rational quotient without identifying its invariant cohomology with the cohomology of a resolution. Let $\Gamma$ be the graph correspondence of $M\dashrightarrow Y$. Choose nonempty opens $U\subset M$, $V\subset Y$ on which it is the finite étale $H$-quotient. For $\alpha\in H^3(M)^H$, set $\beta=\Gamma_*\alpha$. On these opens,

$$ q^*(\beta|_V)=\sum_{h\in H}h^*(\alpha|_U)=32\alpha|_U. \tag{10.13} $$

Remove a divisor supporting $\beta$, its inverse image, and the complements of these opens, enlarging the union to a divisor if necessary. Equation (10.13) shows that $\alpha$ vanishes on the remaining open. Therefore $\mathcal W\subset N^1H^3(M)$.

In weight three, divisorial support is the sum of Gysin images of $H^1(\widetilde D)(-1)$ from resolutions of divisors. This follows from localization and purity of the weight-three piece; strata of codimension at least two contribute no new degree-three support. Compose a finite collection of these Gysin correspondences with the projector onto $\mathcal W$. A finite sum surjects onto $\mathcal W$. The corresponding quotient of the product of Picard varieties, and Poincaré complete reducibility, give an abelian tenfold with (10.2). For each smooth divisor resolution its $H^1$ is algebraically identified with that of its Picard variety by the Poincaré bundle. Curve-to-Prym correspondences replace the two $G$ factors by $P$. This proves the first assertion of Theorem 10.1 for every smooth branch configuration. Appendix B gives an explicit curve, incidence projector and cylinder comparison on a nonempty open of that configuration space.

10.4. Schoen cycles at a compact-type limit

For fixed distinct $a,b_1,\ldots,b_6$, choose distinct constants in each pair $r_j,s_j$, and consider the one-parameter family

$$ C_\varepsilon:\quad w^4=(t-a)^2\prod_{j=4}^6(t-b_j) \prod_{j=1}^3(t-b_j-\varepsilon r_j)^3 (t-b_j-\varepsilon s_j)^2. \tag{10.14} $$

For general nonzero $\varepsilon$ its branch tuple is $1^3,3^3,2^4$. The degree is twenty, so infinity is unramified. The genus is ten, and the three holomorphic character dimensions are $(4,2,4)$. Its primitive Prym $B_\varepsilon$ is an eightfold of Gaussian signature $(4,4)$. The tuple is simple in Schoen's sense: it pairs as three $(1,3)$ and two $(2,2)$ pairs. Consequently [18, Corollary 3.1] supplies both rational Weil classes in $\bigwedge^8_{\mathbb Q(i)}H^1(B_\varepsilon)$ by algebraic cycles.

Blow up the three collisions in the base family and make a finite base change to obtain the admissible cover. On the core the colliding exponents sum to five; dividing $w$ by $\prod_{j=1}^3(t-b_j)$ recovers exactly $G$. A tail has local equation $w_j^4=(x-r_j)^3(x-s_j)^2$, with exponent three at infinity. Its tuple $(3,2,3)$ has genus one and holomorphic deck character $-i$. It is the Gaussian elliptic curve with the inverse of the original deck action on $E$.

At each attachment the two monodromies are one and three, so there is exactly one preimage of the node. The covering dual graph is a tree with a genus-seven core and three elliptic leaves. Thus the limit is of compact type; its Jacobian extends to an abelian scheme with special fiber $J(G)\times E^3$. Applying the primitive idempotent gives, up to equivariant isogeny,

$$ B_0\sim P\times E^3. \tag{10.15} $$

Apply Schoen's construction over the geometric generic point. Its finite collection of cycles descends after a finite extension of the function field. Their closures on the smooth proper abelian scheme specialize to algebraic cycles on $B_0$. Smooth proper specialization identifies the rational cohomology local systems, including both Gaussian determinant eigenlines. Thus the two independent cycle classes remain independent. Pulling through the equivariant isogeny (10.15) gives (10.3). The inverse tail character is essential: the first line has type $(1,4)+(3,0)=(4,4)$.

This construction works for each smooth core, including cores with additional Hodge tensors. It asserts algebraicity of the named space; it does not assert that it generates all new classes at such special points.

10.5. Full-group invariants and algebraic retractions

Assume the full unitary Hodge group. On $W_\sigma$ the Hodge circle has weights $(1,-1,-1,-1,-1)$, and on $e$ it has weight one. The determinant character therefore gives

$$ \operatorname{Hg}(E\times P)_\mathbb C =\{(g,z)\in GL_5\times\mathbb G_m:\det(g)z^3=1\}=:G. \tag{10.16} $$

Indeed the derived group is $SL_5$, and the nontrivial projection to the elliptic Hodge group gives the entire one-dimensional center. The displayed connected group has precisely this derived group and center dimension, proving equality. For the K3 in the theorem, endomorphism field exactly $\mathbb Q(i)$ also gives this full unitary group by the unitary K3 Hodge-group theorem [23]; the finite central isogeny between (10.1) and (10.16) preserves the derived group.

The first fundamental theorem for $SL_5$ generates tensor invariants by contractions and alternating volume tensors. The center of (10.16) forces each volume tensor to be paired with three elliptic characters; the conjugate volume uses the opposite characters. Contractions are polarization classes and endomorphism graphs; the corrected volumes are (10.3). Hence all tensor invariants are algebraic. Rational Künneth and alternating projectors yield HC on every power of $E\times P$. Rational descent causes no extra assertion: the sum of the two conjugate volume lines is the complexification of the rational algebraic space already constructed. Base change of its rational cycle-class span proves the assertion about rational invariants.

Let $u$ denote the correspondence in (10.2). Its polarization adjoint is algebraic: inverse Lefschetz operators on abelian varieties are algebraic by Lieberman's theorem [24], and ordinary polarizations and transposes of correspondences are algebraic. The positive operator $u^\dagger u$ is invertible; by Cayley–Hamilton its inverse is a rational polynomial in itself. Thus $(u^\dagger u)^{-1}u^\dagger$ is an algebraic left inverse to $u$.

All cohomology of $C^n$ is consequently an algebraic direct summand of a finite sum of Tate twists of cohomology of powers of $E\times P$. Use the exterior-algebra description of abelian cohomology and its algebraic projectors. Algebraic embeddings and retractions transfer HC in both directions needed here. The product-map graph in §9.3 similarly embeds $T(S_f)$ into $H^2(E\times P)$ with an algebraic retraction; the other K3 cohomology is Tate. This proves ordinary HC for all $S_f^r\times C^n$. Buskin's algebraic comparison transfers it to the other K3 surfaces in the theorem.

10.6. Fully twisted sources and actual support

For $1\le k\le4$, retain the entire central character in the weight-one complex modules

$$ F_k=e^{k-1}\otimes\bigwedge^k W_\sigma\,(k-1). \tag{10.17} $$

Their Hodge signatures are $(1,4),(4,6),(6,4),(4,1)$. The first two conjugate pairs are $H^1(P)$ and $H^1(A)$. The determinant identifies the last two with the conjugates of the first two, with the Tate weights as in (10.17). Thus every fundamental module has an actual abelian source in $C$.

Every complex irreducible $G$-module is $e^b\otimes\mathbb S_\lambda W_\sigma$, with $\lambda=(\lambda_1,\lambda_2,\lambda_3,\lambda_4,0)$ dominant; determinant powers are absorbed using $\det W_\sigma=e^{-3}$. Write $\lambda=\sum_{k=1}^4a_k\omega_k$, and set

$$ m=\sum_k a_k=\lambda_1,qquad d=b-\sum_k(k-1)a_k. \tag{10.18} $$

The exponent of the sole positive circle coordinate in a weight of $\mathbb S_\lambda W_\sigma$ ranges from zero to $\lambda_1$, and Weyl-orbit weights attain both endpoints. The circle extrema are therefore $d-m$ and $d+m$. Including conjugation, its rational Hodge level is

$$ \ell=m+|d|. \tag{10.19} $$

The Cartan component of a tensor of $a_k$ copies of $F_k$ has weight $m$ and the required derived-group highest weight. Adding $|d|$ elliptic characters of the appropriate sign realizes the given module at weight $\ell$, with a nonzero extreme Hodge component. Use distinct factors to place this tensor in ordinary cohomology of some $C^N$.

For clarity, rational descent can be stated without choosing a complex line as a rational source. Let $U\subset H^j(X,\mathbb Q)$ be an irreducible rational Hodge structure for $X=S_f^r\times C^n$, and set $p=(j-\ell)/2$. The rational source $M=H^\ell(C^N,\mathbb Q)$ just constructed and $U(p)$ share a complex irreducible constituent. Reductivity and base change for equivariant Hom give

$$ \operatorname{Hom}_G(M,U(p))\otimes\mathbb C =\operatorname{Hom}_{G_\mathbb C}(M_\mathbb C,U(p)_\mathbb C)\ne0. \tag{10.20} $$

A nonzero rational map is surjective since $U$ is irreducible. Equal weight and equal circle action make it a Hodge morphism. This argument handles rational multiplicities and Schur indices.

Ordinary HC on $C^N\times X$, proved above, represents this surjection by a cycle of codimension $\dim C^N+p$. The projection of its support to $X$ has dimension at most $\dim X-p$. The correspondence therefore supports $U$ in codimension at least $p$. Summing irreducible summands proves generalized HC. This is the actual cycle-dimension argument underlying the domination strategy of [21] and [22].

10.7. The prescribed even-Clifford correspondence

For $R_\mathbb Q=I_{2,8}$ the even Clifford algebra has center $K=\mathbb Q(i)$ and is split over its center: after adjoining $i$ the quadratic form is hyperbolic. Thus its minimal rational spin module has dimension thirty-two. The central character can be checked explicitly in the restriction to (10.16). Put $V_\sigma=eW_\sigma$; then $\det V_\sigma=e^2$. One complex half-spin representation is

$$ \begin{aligned} \det(V_\sigma)^{-1/2} \left(\bigwedge^0 V_\sigma\oplus\bigwedge^2V_\sigma \oplus\bigwedge^4 V_\sigma\right) &=e^{-1}\oplus e\bigwedge^2W_\sigma \oplus e^3\bigwedge^4W_\sigma\\ &=e^{-1}\oplus e\bigwedge^2W_\sigma\oplus W_\sigma^*. \end{aligned} \tag{10.21} $$

With its conjugate and the weight-one normalization, this is exactly $H^1(E\times P\times A)$, in agreement with [17]. The regular even-Clifford Hodge structure is a sum of copies of this minimal module. Hence its Kuga–Satake abelian variety is isogenous to a power of $C$.

Every specified rational tensor defining the standard Kuga–Satake map is now a Hodge class on a mixed product of the K3 and this abelian variety. Ordinary mixed HC proves its algebraicity, including the determinant-sensitive middle blocks. Isogeny correspondences also transfer generalized HC to all the stated mixed products. This proves the full-group part of Theorem 10.1. The precise object here is the Kuga–Satake variety of the rank-ten transcendental form; no choice of a larger primitive lattice is being silently substituted.

10.8. The prescribed class at special Gaussian points

We now remove the full-group hypothesis for ordinary HC and for algebraicity of the prescribed KS tensor. Section10.9 then treats generalized support by the actual smaller group. The specialization argument is a reuse of the relative-class method in our earlier paper [24, §§9.5–9.6]; the additional transfer of ordinary HC uses Floccari [25, Theorem 3.5].

Around any smooth seven-point configuration, the K3 models have the same six type-III and one type-$I_0^*$ fibers. A finite base change and simultaneous resolution of their rational double points give a smooth projective polarized K3 family. Fix a smooth algebraic curve in the configuration base through the desired parameter and a very general full-group parameter, removing its finitely many bad points. A further finite cover supplies the family and the persistent divisor marking. The rank-ten orthogonal complement $\mathcal V$ of those divisors is a polarizable variation even when the transcendental rank jumps. Because the curve contains a full-group point, its very general point has the full group as well.

The relative KS construction [5, Chapter 6, §4.4 and Proposition 4.10] gives, after a finite étale cover, an algebraic abelian scheme with weight-one fibers $C^+(\mathcal V_b)$. Its monodromy is a Spin lift acting by left multiplication on the reference Clifford algebra. It is not the naive Clifford functor applied to orthogonal monodromy; that would not in general give the required global polarization. Fix a nonisotropic reference vector $v_0$ as right Clifford datum. For $\phi_v(w)=vwv_0$,

$$ L_g\phi_vL_g^{-1}(w)=(gvg^{-1})wv_0. \tag{10.22} $$

Thus the specified embedding is monodromy-equivariant even though $v_0$ need not be a flat vector of $\mathcal V$. Using the global abelian polarization gives a flat rational codimension-two KS class $\kappa$ on the relative product of the K3 and two copies of the abelian scheme. Section 10.7 makes it algebraic at very general points.

For completeness, a fixed flat class algebraic at very general points of a smooth projective family over a complex curve is algebraic at every point. Relative Hilbert schemes have countably many components; finite products and rational coefficient lists parameterize rational cycle combinations. Their maps to the base are proper. Choose a point outside the images of all nondominating components and outside the exceptional set of the very-general assertion. A representing cycle combination belongs to a dominating component. Its universal cycle has the specified flat class throughout that connected component. A finite multisection, followed by normalization and the valuative criterion of properness, specializes it to every base point. Equally, the proper dominating component itself surjects onto the base. This applies to $\kappa$. It does not apply to arbitrary new Hodge classes appearing only at a special point.

We also record the extraction of the actual smaller transcendental KS class. At a special point write $V=T\perp N$, with $N$ Tate, and choose nonisotropic $t_0\in T$. Right multiplication by the even invertible element $v_0t_0/q(v_0)$ commutes with the left KS complex structure, hence is an algebraic abelian quasi-isogeny. Composing on the output changes $\phi^{v_0}$ to $\phi^{t_0}$. In the graded Clifford decomposition choose inclusion $i:C^+(T)\to C^+(V)$ and projection $p$ discarding each nonempty $N$-monomial. They commute with the left KS complex structure in $C^+(T)_\mathbb R$, hence are actual rational abelian homomorphisms and $pi=1$. For $v\in T$,

$$ p\phi_V^{t_0}(v)i(w)=vwt_0=\phi_T^{t_0}(v)(w). \tag{10.23} $$

The source projector onto $T$ is algebraic by the inverse intersection matrix of a divisor basis. Consequently the specialized cycle extracts the prescribed smaller KS class, rather than an unspecified nonzero embedding. These operations are the choice changes of [5, Chapter 4, Proposition 2.6 and Remark 2.7]. Adjoining a Tate complement gives isogeny powers of the same KS variety [5, Chapter 4, Example 2.4].

For every smooth source, the order-four action restricts to an endomorphism of $T(S_f)$ squaring to minus one. Its endomorphism field is therefore CM. By [14, Corollary 2.3], HC holds on every power of $S_f$. Floccari's theorem [25, Theorem 3.5] states that, once the prescribed KS correspondence is algebraic, HC on all K3 powers is equivalent to HC on all powers of its KS variety. It follows that HC holds for all KS powers at every smooth source.

The spin restriction (10.21) is an identity of representations and Hodge circles and remains valid when its factors become reducible. Thus $C_f=E\times P_f\times A_f$ is an isogeny factor of a power of the KS variety, also at special points. HC for all $C_f^n$ follows, and the algebraic K3 source/retraction gives HC for all mixed $S_f^r\times C_f^n$. Isogeny transport in §10.1 proves the ordinary HC conclusion for every Gaussian fivefold in Corollary 10.2. For the other K3 surfaces of Proposition 9.1, Buskin's comparison and the KS functoriality in [25, §3.5 and Remark 3.7] transfer the prescribed correspondence and ordinary mixed HC, including at Picard jumps. This strengthening relies on the specified flat class and the two cited Hodge theorems; it is not a specialization principle for all Hodge classes.

10.9. Generalized support at Hodge-group drops

We prove generalized HC at every smooth Gaussian source by calculating its actual smaller Hodge group. This is a direct domination argument, not specialization of generalized HC. The use of all spin constituents is essential, including both half-spin representations.

Let $T$ be the actual transcendental space, with CM endomorphism field $F\supset K=\mathbb Q(i)$. Put $d=[F^+:\mathbb Q]$ and $m=\dim_FT$, so $dm\le5$. Since $F^+$ is totally real,

$$ F=F^+K. \tag{10.24} $$

Zarhin's theorem [23], also [5, Chapter 3, Theorem 3.9], gives $\operatorname{Hg}(T)=\operatorname{Res}_{F^+/\mathbb Q}U(T,h)$. Over $\mathbb C$ it is $\prod_{\alpha=1}^dGL(U_\alpha)$, where $\dim U_\alpha=m$. Choose the embeddings of $F$ extending one fixed embedding of $K$. Their sum is an isotropic half $U=\bigoplus U_\alpha$ of $T_\mathbb C=U\oplus U^*$. The Hodge circle has weight two at one coordinate of the selected factor, and zero at all other coordinates of $U$.

The connected inverse image of this group in the spin group is

$$ \widetilde G_\mathbb C= \left\{((g_\alpha),z):\prod_\alpha\det(g_\alpha)=z^2\right\}. \tag{10.25} $$

Write $e$ for its character $z$. The preimage is connected: on a maximal torus the defining character has coefficients including one, so its kernel is a torus, and the derived factors $SL_m$ are connected. The actual KS Hodge group projects onto $\operatorname{Hg}(T)$, hence has the same dimension as this connected inverse image and therefore equals it. In particular its lifted circle sends $-1$ to the central $-1$, as seen from its weight-one action. We are discussing Hodge groups; the weight scalar in the Mumford–Tate group is separate.

The full spin module restricts to

$$ e^{-1}\bigwedge^{\bullet}U =\bigoplus_{0\le k_\alpha\le m} e^{-1}\bigotimes_\alpha\bigwedge^{k_\alpha}U_\alpha. \tag{10.26} $$

Both complex half-spin modules occur in the left regular $C^+(T)$ because $\dim_\mathbb Q T$ is even. Thus every summand in (10.26) occurs in the actual $H^1$ of its KS variety. At any Galois-selected active factor the summand has only circle weights $\pm1$: $k_\alpha=0$ gives $-1$, $k_\alpha=m$ gives $+1$, and an intermediate exterior power gives both. Tate complements and abelian isogenies place these sources in the mixed category already covered by ordinary HC in §10.8.

The Galois action on the half-determinant requires care. By (10.24), Galois automorphisms fixing $K$ permute the $U_\alpha$; conjugating $K$ dualizes all of them at once. Since $2e=\sum_\alpha\det(U_\alpha)$ in additive character notation, and a connected torus has torsion-free character lattice, these operations respectively fix $e$ or replace it by $-e$. Inner forms and multiplicities do not change this character calculation. There is no independent sign change on a subset of the factors.

A complex irreducible representation of (10.25) has the form

$$ R=e^b\bigotimes_\alpha\mathbb S_{\lambda_\alpha}U_\alpha, \qquad b\in\mathbb Z,\quad \lambda_{\alpha,1}\ge\cdots\ge\lambda_{\alpha,m}\in\mathbb Z. \tag{10.27} $$

Negative entries are allowed. For the circle active at factor $\alpha$, its extrema are $b+2\lambda_{\alpha,m}$ and $b+2\lambda_{\alpha,1}$. The real embeddings are Galois-transitive, and conjugation reverses the extrema. Consequently the rational irreducible Hodge structure containing $R$ has level

$$ \ell=\max_\alpha\max\left( |b+2\lambda_{\alpha,m}|, |b+2\lambda_{\alpha,1}|\right). \tag{10.28} $$

All extrema have parity $b$. Set $c=(b+\ell)/2\in\mathbb Z$ and $\lambda'_\alpha=\lambda_\alpha+(c,\ldots,c)$. Then

$$ 0\le\lambda'_{\alpha,j}\le\ell,\qquad R=e^{-\ell}\bigotimes_\alpha \mathbb S_{\lambda'_\alpha}U_\alpha. \tag{10.29} $$

Each partition $\lambda'_\alpha$ fits in an $m$-by-$\ell$ rectangle. Write it as $\ell$ columns, permitting heights zero and $m$, and align those columns across $\alpha$. Each aligned column is an actual weight-one spin summand in (10.26). The Cartan component of the tensor product of the $\ell$ columns is exactly (10.29). It lies in $H^\ell(\operatorname{KS}(T)^\ell,\mathbb C)$ and has an extreme $(\ell,0)$ or $(0,\ell)$ component; its rational orbit is fully twisted. If $\ell=0$, the representation is trivial and the source is a point.

For a rational irreducible substructure of degree $j$ in a fixed-source mixed product, use the rational Hom descent in (10.20) with this source. It gives a rational Hodge surjection at weight $\ell$. The ordinary mixed HC theorem of §10.8 represents it by a cycle, whose projection has codimension at least $(j-\ell)/2$. Decomposing arbitrary substructures proves generalized HC at every smooth Gaussian source. The same isogeny and K3 comparisons used for ordinary HC transfer this statement to Corollary 10.2 and the K3s in Proposition 9.1. This completes Theorem 10.1 without a full-group restriction.

The calculation includes CM points ($m=1$), lower-rank points with endomorphism field $K$, and the non-CM case $d=m=2$. It does not rely on all real derived factors being noncompact. It treats powers and mixed products attached to one source at a time, and makes no assertion about arbitrary tuples of unrelated special sources.

10.10. Scope of the advance

The cycle constructions in §§10.2–10.4 and the isogeny transport in §10.1 require no generic-Hodge-group assumption. Ordinary HC and the prescribed KS correspondence hold also at special points by Section10.8. Section10.9 proves generalized support there by calculating the actual smaller group and its full spin representation. All mixed statements concern a fixed source and its associated varieties; arbitrary tuples of unrelated special sources are not covered. Neither a generic $SO(2,8)$ source nor a deformation of these cycles in the four normal directions is constructed here. In the original surface family, §9.6 transfers the selected Gaussian cohomology; we do not claim full HC for arbitrary powers of the entire original surface from that selected summand alone.

The proof gives actual correspondences by support, specialization and isogeny. Appendix B additionally describes the tenfold as an incidence image in the Jacobian of an explicit line curve on a nonempty open. That geometry does not yet provide the normal deformation of the full spin source. The general rational Hodge conjecture remains open.

11. Reproducibility, credit and unresolved work

The supplement contains eleven Python scripts using SymPy and python-flint. Run them from its root:

python code/check_seed13_rank16_Kummer_lattice.py
python code/check_seed13_Ma12_K3_model.py
python code/check_seed13_order4_Ma_source.py
python code/check_Gaussian_tenfold_quotient.py
python code/check_Gaussian_Schoen_degeneration.py
python code/check_Gaussian_spin_rectangle_support.py
python code/check_Gaussian_two_torsion_resolution.py
python code/check_Gaussian_Eckardt_global_comparison.py
python code/check_Gaussian_cubic_fibration_boundary.py
python code/check_Eckardt_line_projector.py
python code/explore_Gaussian_Eckardt_conic.py

They write JSON certificates to the results directory. The first checks all affine hyperplanes in $\mathbb F_2^4$, the relevant index and parity relations, explicit root-lattice gluing and the local $A_3$ matrices. The second checks the explicit quadric model and both elliptic fibrations. The third checks the Gaussian subfamily, its smooth sample and the actual cyclic product map. The fourth checks the quotient field identities, the order-32 group, its invariant Hodge numbers and the canonical degree. The fifth checks the balanced smoothing, tail characters, compact-type graph and central-character level formula. The sixth checks the rectangular spin source construction at every allowed dimension factorization. The next four check the ordered-resolution algebra and defect, Gaussian quotient comparison identities, the cubic-pencil boundary, and the twenty-seven-line incidence projector. The final script supplies an exact conic equation certificate. The ten checker scripts have 25,25,21,15,22,5,14,14,6 and8 passing summary checks, totaling155; these include3328 bounded central-level and30,632 rectangular-partition controls. The support, specialization and density arguments are written proofs; finite checks do not establish them by themselves. They require no GPU or external data. Their success is evidence for these finite consequences; the primitive Kummer closure theorem, local geometric identifications and imported comparison theorems are mathematical inputs proved or cited in the text.

The full period-rank proof combines lelong_ridge's type-$(1,3)$ seed and rank proposal, gabrielov_cell's cyclic-three block decomposition and normal map, and our global logarithmic identification, residue audit and theta-avoidance argument. gabrielov_cell also described the actual double-Kummer cover and supplied (8.1). The exact determinant and primitive saturation, odd-unimodular identification, boundary synthesis and conditional comparison are developed here. Discussion replication of finite computations is not described as an independent geometric proof. The relevant discussion is preserved at [27]. The new Section10 quotient, compact-type specialization, isogeny transport and mixed support are developed by cm_symmetry_scout, building on the cited literature. The coauthors' contributions to the earlier geometric comparison remain as specified above; no independent endorsement of Section10 or Appendix B is implied.

The remaining generic problem is concrete: construct an actual algebraic correspondence from a tensor of curve $H^1$ spaces, or an abelian variety, carrying the eight-dimensional orthogonal variation with rank-ten fibers. Sections 9–10 supply such a source, every Gaussian spin factor, and mixed Hodge conjectures on the four-dimensional Gaussian sublocus of the K3 target, independently of the conditional comparison. Section 9.6 makes its conditional transfer precise on the open of original surfaces reached by the full-base spread. Rationality of the moduli space, the presence of the genus-four fixed curve, and the comparison (1.2) do not provide that correspondence. The original rational Hodge conjecture remains unresolved.

Appendix A. Full period rank of the type-$(1,3)$ family

We include the proof because a numerical Jacobian rank, without identification with the actual period derivative, would not suffice. The six-dimensional residue block adapts [1, §3.4], keeping an order-three line bundle and its inverse distinct. The cyclic-point formulation and block decomposition were developed in discussion with lelong_ridge and gabrielov_cell; the global quotient, theta-avoidance argument and complete synthesis below make the geometric rank assertion explicit.

A.1. Global logarithmic quotient and the actual derivative

For a double cover of an abelian surface with branch $D=(d=0)$, put $E=T_B(-\log D)$ and fix a nonzero invariant volume form. Rank-two wedge duality gives

$$ E\otimes L\simeq\Omega_B^1(\log D)\otimes L^{-1}. \tag{A.1} $$

The anti-sheet Hodge space $H^{1,1}$ is $H^1$ of the right-hand side. The logarithmic normal sequence tensored by $L$ is

$$ 0\longrightarrow E L\longrightarrow T_B L \longrightarrow L^3|_D\longrightarrow0. \tag{A.2} $$

Ample vanishing and the restriction sequence identify

$$ H^{1,1}_-(X)\simeq \frac{H^0(B,L^3)} {dH^0(B,L)+\text{lifted normal derivatives of }H^0(B,T_BL)}. \tag{A.3} $$

There is no assumption here that a collection of meromorphic exact forms exhausts a kernel. The map on $H^0(T_BL)$ is injective: a section tangent to $D$ restricts to a section of $T_D\otimes L|_D=L^{-1}|_D$, which has negative degree. It must vanish along $D$ and then lie in $H^0(T_BL^{-1})=0$.

A branch direction $e\in H^0(L^2)/\langle d\rangle$ has Kodaira–Spencer class given by the connecting map of the untwisted logarithmic normal sequence. Multiplication by a theta section $\theta\in H^0(L)$ gives a commutative diagram with (A.2). Naturality of connecting maps identifies the period derivative of the form $\theta\,\Omega/z$ with $\theta e$ in (A.3), up to a universal nonzero normalizing scalar. Locally a vector field with $v(d)=a d$ lifts as $v+(a/2)z\partial_z$, which gives the same contraction formula.

For invariant vector fields $\partial_i$ and $f\in H^0(L)$, representatives of the normal derivatives are

$$ J_i(f)=d\,\partial_i f-\frac12 f\,\partial_i d. \tag{A.4} $$

These are global sections of $L^3$: derivative-of-frame terms cancel under every line-bundle frame change. The formulas commute with finite automorphisms and their rational character projectors.

A.2. The cyclic-three point

Choose an indecomposable principally polarized abelian surface $(B_0,M)$, a nontrivial order-three bundle $\chi$, and the étale cyclic isogeny $\pi:B\to B_0$ trivializing $\chi$. Put $L=\pi^*M$ and $d=\pi^*d_0$, with $d_0\in H^0(M^2)$ smooth. Inversion conjugates the order-three deck transformation to its inverse. The three theta lines have characters $1,\chi,\chi^{-1}$. The selected holomorphic form has numerator $\theta_\chi-j^*\theta_\chi$.

At this point the dihedral isotypic decomposition splits the selected $U_Y$ into a rank-eight K3-type piece, of types $(1,6,1)$, and a rank-two algebraic piece. The latter is the deck-invariant part of the selected $H^{1,1}$. The six tangent directions preserving cyclic branch structure consist of three polarized abelian deformations and three branch directions. The other four branch directions have the form $s+j^*s$, for $s\in H^0(M^2\chi^{-1})$.

A.3. Rank six in the tangential block

Let $F=(\theta_\chi=0)$ and $K=K_F$. This is a smooth genus-two theta translate, with $\operatorname{Jac}(F)\simeq B_0$. In particular $\eta=\chi^{-2}|_F$ is nontrivial of order three. Adjunction gives

$$ (M\chi)|_F=K,\qquad M^2|_F=K^2\eta. $$

The tangent period map is multiplication by $\theta_\chi$ on $H^1(T_{B_0}(-\log D_0))$. It is injective if the cokernel sheaf of $E\to E M\chi$ has no global section. Upon restriction to $F$, the relevant normal derivative map is

$$ A_d:H^0(K)^{\oplus2}\longrightarrow H^0\bigl(Z,(K^3\eta)|_Z\bigr), \qquad Z=D_0\cap F, \tag{A.5} $$

between spaces of dimension four.

The restriction map $H^0(M^2)\to H^0(F,K^2\eta)$ is surjective with kernel $\mathbb C\theta_\chi\theta_{\chi^{-1}}$. Choose a restriction $s=d|_F$ with four simple zeros avoiding the zeros of $t_0=\theta_{\chi^{-1}}|_F\in H^0(K\eta)$. Such choices exist because $K^2\eta$ is basepoint-free. Replace $d$ by $d+\lambda\theta_\chi\theta_{\chi^{-1}}$. Then $A_d$ changes to $A_d+\lambda A'$. The derivatives of $\theta_\chi$ in two independent translation directions form a basis $\theta_1,\theta_2$ of $H^0(K)$: a nonzero kernel vector would give a translation tangent to an ample theta curve. Thus

$$ A'(a,b)=t_0(a\theta_1+b\theta_2)|_Z. $$

The canonical pencil gives $\operatorname{Sym}^2H^0(K)=H^0(K^2)$. Restriction to $Z$ is injective since $K^2(-Z)=\eta^{-1}$ has no section. Therefore $A'$ has rank three and kernel the skew-gradient vector $k=(\theta_2,-\theta_1)$. It is a nowhere-zero section of $T_F\otimes K=\mathcal O_F$.

For a nonzero constant $c$, $A_d(k)=c\,d_Fs|_Z$. In

$$ 0\longrightarrow\eta^{-1} \xrightarrow{\ s\ }K^2\longrightarrow K^2|_Z \longrightarrow0, $$

the boundary of $(d_Fs)/t_0|_Z$ pairs with $t_0\in H^0(K\eta)$ by Serre duality. Each simple zero contributes $\operatorname{Res}(ds/s)=1$, so the pairing is $4$. It is nonzero. Hence $A_d(k)$ is outside $\operatorname{im}A'$, and the coefficient of $\lambda^3$ in $\det(A_d+\lambda A')$ is nonzero. General $\lambda$ makes (A.5) invertible.

Finally $H^1(E)$ is the six-dimensional polarized pair-deformation tangent space. In the untwisted logarithmic normal sequence, $h^0(K_{D_0})=5$, $h^0(T_{B_0})=2$ and $h^1(T_{B_0})=4$; contraction with $c_1(D_0)$ has one-dimensional image. The quotient of embedded deformations by translations has dimension three, and the polarized abelian deformations have dimension three. The finite isogeny and torsion choices introduce no infinitesimal parameters. This proves rank six in the tangential block.

A.4. Rank two in the normal block

The deck-invariant odd numerator is $H^0(M^3)^-$, of dimension four. By (A.3) its denominator is

$$ J=\langle J_1(\theta_M),J_2(\theta_M)\rangle, \qquad\dim J=2. $$

The denominator has no $dH^0(M)$ term in this parity. The two displayed sections are independent by the injectivity in A.1. The projection of the actual normal derivative is

$$ \Pi(s)=\theta_\chi s-j^*(\theta_\chi s), \qquad s\in H^0(M^2\chi^{-1}). \tag{A.6} $$

The two theta translates are distinct irreducible divisors. If $\Pi(s)=0$, divisibility forces $\theta_{\chi^{-1}}$ to divide $s$; the remaining factor is a multiple of the even $\theta_M$. Conversely this product is in the kernel. Thus $\ker\Pi$ has dimension one and $\operatorname{im}\Pi$ has dimension three.

Let $F_\chi\cap F_{\chi^{-1}}=\{p,-p\}$. The image of $\Pi$ vanishes at these points. One can require transverse intersection, $p\notin B_0[2]$ and $\theta_M(p)\ne0$. To see nonemptiness, degenerate to $E_1\times E_2$ with its product theta divisor, and choose the order-three translation $(a,b)$ with both coordinates nonzero. The translated theta divisors intersect transversely at $(a,-b)$ and $(-a,b)$, neither on the product theta divisor. These open conditions persist on nearby indecomposable abelian surfaces.

On $\theta_M\ne0$,

$$ \frac{J_i(\theta_M)}{\theta_M^3} =-\frac12\partial_i(d/\theta_M^2). $$

The $|2M|$ Kummer morphism is immersive off $B_0[2]$. For general $d$ at least one of these derivatives is nonzero at $p$. Therefore $J$ is not contained in $\operatorname{im}\Pi$. Since their dimensions are two and three in a four-dimensional space, $J+\operatorname{im}\Pi=H^0(M^3)^-$. The normal block surjects onto the two-dimensional quotient by $J$.

The rank-six tangential block maps into the complementary isotypic space, so the full period derivative has rank $6+2=8$. All the conditions used are nonempty opens of the same irreducible section and level family, so they can be imposed simultaneously. The rank condition is algebraic on Hodge bundles; it holds on a nonempty Zariski open of the type-$(1,3)$ family.

The K3-type period domain for rank ten has dimension eight, so the image contains an analytic open set. Outside countably many proper rational tensor Hodge loci the Hodge group is the full $\operatorname{SO}(U_Y)$: the Hodge circles rotating positive two-planes in an open period ball generate the connected real orthogonal group. A proper rational subgroup would impose a proper rational tensor Hodge locus. The generic Hodge structure is consequently simple and has no rational $(1,1)$-class.

Appendix B. An explicit curve and projector for the Gaussian tenfold

The support construction in §10.3 works at every smooth branch configuration. On a nonempty open of that configuration space, we can also specify the supporting curve and identify its cylinder with the particular invariant space $\mathcal W$ in (10.12). This uses classical cubic-surface incidence [27, §§1.1–1.13 and 9.1]. The additional point here is the explicit Gaussian family and its comparison with the specified exterior-square source; the general Prym–Tyurin construction is not new.

Send the double branch point $a$ to infinity. On $\mathbb P^2_{[Z:S:P]}$ put

$$ L_i=P-b_iS+b_i^2Z,\qquad A_3=L_1L_2L_3,\qquad B_3=L_4L_5L_6. \tag{B.1} $$

The seven lines $Z,L_1,\ldots,L_6$ are distinct tangents to a smooth conic, so no three concur. The quotient threefold of §10.2 is birational to

$$ \mathcal Z:\quad w^2=(Zt^2)^2-4A_3B_3 \subset\mathbb P(1,1,1,1,3). \tag{B.2} $$

For example, in the affine base chart $Z=1$, its Weierstrass coordinates $x,y$ give $t=y/x$ and $w=2x-t^2$.

Proposition B.1. On a nonempty open of the six finite branch-point configuration, there is a smooth projective threefold $X$ birational to (B.2), a morphism $f:X\to\mathbb P^1$, and an involution $\iota:t\mapsto-t$ over the base with

$$ \begin{aligned} H^3(X,\mathbb Q)^+&\simeq H^1(T,\mathbb Q)(-1),\qquad g(T)=2,\\ H^3(X,\mathbb Q)^-&\simeq\mathcal W. \end{aligned} \tag{B.3} $$

Both isomorphisms are induced by algebraic correspondences. If $\mathcal C_{27}$ is the normalized curve of lines of its smooth cubic fibers and $D$ is the incidence correspondence sending a line to the ten lines meeting it, then

$$ e_{\mathrm{line}}=\frac{(1-D)(1-\iota)}{12},\qquad A_{\mathrm{line}}=\operatorname{im} \bigl(e_{\mathrm{line}}:J(\mathcal C_{27})\to J(\mathcal C_{27})\bigr) \tag{B.4} $$

define, up to isogeny, an abelian tenfold and the isomorphism (10.2). The Jacobian of a disconnected normalization means the product of its component Jacobians. Only the twenty-four non-Eckardt lines contribute to (B.4).

B.1. A global small resolution, including the Gaussian fiber

Write $R=Zt^2$, $u=(R+w)/2$ and $v=(R-w)/2$. Thus

$$ uv=\prod_{i=1}^6L_i,\qquad u+v=R. \tag{B.5} $$

Blow up, in the stated order, the proper transforms of the Weil divisors $(u,L_1),\ldots,(u,L_5)$. We verify smoothness, including the central Gaussian singularities, before using a deformation argument.

The singularities away from $p=[0:0:0:1]$ are precisely the fifteen points $t=L_i=L_j=0$. Indeed, a singular point has $w=0$. If $H=\prod_iL_i\ne0$, then $R\ne0$, so $Z,t\ne0$ and the nonzero $t$-derivative $4Z^2t^3$ excludes a singularity. If $H=R=0$, the condition $dH=0$ requires at least two vanishing $L_i$. Their intersection line meets $R=0$ at $p$ and at the double zero $t=0$.

At the latter point use $x=L_i$, $y=L_j$ and $H=xyh(x,y)$ with $h(0,0)\ne0$; also $Z=\ell(x,y)$ is a unit. The local singularity is $uv=xyh$, $u+v=\ell t^2$, of type $cA_3$. The first ordered blowup meeting either factor has charts

$$ \lambda\ell t^2-x\lambda^2-yh=0, \qquad x-\alpha\ell t^2+\alpha^2yh=0. \tag{B.6} $$

Along the exceptional curve their respective $y$- and $x$-derivatives are $-h(0,0)$ and $1$. They are smooth. The remaining factor ideals are principal there by the transformed equation, so later blowups are locally trivial.

At $p$, set $t=1$, and put $A_k=\prod_{j\le k}L_j$, $B_k=\prod_{j>k}L_j$. After $k$ main charts the equations are

$$ u_kv=B_k,\qquad A_ku_k+v=R. \tag{B.7} $$

The next complementary chart is

$$ L_{k+1}-u_k\alpha=0,\qquad R-A_ku_k-\alpha B_{k+1}=0. \tag{B.8} $$

For $k=0$, eliminate $u_0$; the remaining gradient at $p$ is $L_1-\alpha Z\ne0$. For $1\le k\le4$, above $p$ we have $u_k\alpha=0$. If $u_k=0$, the second gradient in the three base coordinates is $Z$, except for $k=4$, when it is $Z-\alpha L_6$. If $\alpha=0$, it is $Z$, except for $k=1$, when it is $Z-u_1L_1$. The independence of every triple $Z,L_i,L_j$ makes the two equation gradients independent, including their common origin. In the final main chart the equation is

$$ u_5R-A_5u_5^2-L_6=0, \tag{B.9} $$

whose gradient at $p$ is $u_5Z-L_6\ne0$. The exceptional locus is one rational curve over each $cA_3$ point and a chain of five over $p$. These projective blowups therefore give a smooth projective small resolution $X$.

The same charts are algebraic in the six plane equations and the cubic $R$. Their fiber Jacobians have maximal rank at every point of the resolved Gaussian fiber. Properness and the relative Jacobian criterion give a smooth projective deformation to general six planes in $\mathbb P^3$ and a general cubic $R$. The general double sextic has forty-five nodes $R=L_i=L_j=0$ and no others. The union of its fifteen pair-intersection lines has ideal generated by $H/L_i$ and resolution

$$ 0\longrightarrow S(-6)^5\longrightarrow S(-5)^6 \longrightarrow I_{\mathrm{lines}}\longrightarrow0, \qquad S=\mathbb C[z_0,z_1,z_2,t]. \tag{B.10} $$

The coordinate ring is Cohen–Macaulay and $R$ is a nonzerodivisor; thus the node ideal $I_{\mathrm{lines}}+(R)$ is saturated. Its degree-five piece has dimension $6+10=16$. The two summands have zero intersection: if $Rq$ vanishes on each line, $q$ does too, whereas the line ideal has no degree-two element. The forty-five nodes therefore impose forty conditions on quintics, so the defect is five by the double-cover formula [28, Proposition 3.7 and the proof of Theorem 4.6]. A smooth double sextic has $h^{1,1}=1$, $h^{2,1}=52$, from the degree-five weighted Jacobian ring. The small-resolution transition gives

$$ h^{1,1}(X)=6,\qquad h^{2,1}(X)=52-45+5=12, \qquad b_3(X)=24. \tag{B.11} $$

These Hodge numbers transfer to the Gaussian fiber by the smooth projective deformation just verified.

The function $r=u/A_3$ extends to a morphism $f:X\to\mathbb P^1$. After three main charts, $[u:A_3]=[u_3:1]$. In an earlier complementary chart, with $k=0,1,2$, it is $[1:\alpha\prod_{j=k+2}^3L_j]$; neither expression has a base point. The last two blowups preserve the map. Over $r\ne0,\infty$, $u_3=r$ is a unit, those last blowups are trivial, and eliminating $v$ gives exactly the cubic-surface family

$$ F_r:\quad A_3r^2+B_3-Zrt^2=0\subset\mathbb P^3. \tag{B.12} $$

All ordered blowup ideals are preserved by $t\mapsto-t$.

B.2. The two line cylinders fill the third cohomology

The smooth fibers of (B.12) have Eckardt point $[0:0:0:1]$. The involution fixes this point and the plane elliptic section $t=0$. Lefschetz gives $H^2$ eigendimensions three and four. Removing the canonical class leaves local systems $L^+$ and $L^-$ of ranks two and four.

The plane-cubic pencil $B_3+\lambda A_3$ has two triangle fibers, six other simple nodal fibers, and four simple tangencies to $Z=0$, with these parameter sets disjoint, on a nonempty open. The supplement certifies that open at $(b_1,\ldots,b_6)=(0,1,3,7,11,19)$ modulo $431$: the residual node and tangency parameter polynomials have respective degrees six and four, are square-free and coprime, and the node Hessians are units. Both charts at binary infinity are checked. The pencil has nine transverse base points, so their blowup is a rational elliptic surface of Euler number twelve. The two triangles contribute three each; hence the six nonzero nodal parameters contain exactly one node apiece and exhaust the remaining singular plane-cubic fibers. These are algebraic open conditions, so the certificate proves nonemptiness in characteristic zero.

After $\lambda=r^2$, there are twelve single-node cubic-surface fibers and eight fibers with two nodes exchanged by $\iota$. At a single node, $\iota$ reverses one coordinate of the local $A_1$ equation and acts by $-1$ on its vanishing sphere. The reflection has invariant codimension one on $L^-$ and zero on $L^+$. At an exchanged pair the two orthogonal vanishing classes give one plus and one minus combination; the invariant codimension is one on each system.

At $r=0$, the change $t'=\sqrt r\,t$ gives $B_3+r^2A_3-Zt'^2=0$. The central cubic has three $A_1$ points. The $r^2$ pullback squares their commuting reflections, killing that monodromy; the remaining twist around the $r$-loop is $t'\mapsto-t'$, namely $\iota$. Thus the endpoint monodromy on $L^-$ is $-1$ and on $L^+$ is $+1$. The same holds at infinity. This is a local-system calculation; an equivariant simultaneous small resolution at the endpoint is not assumed.

Let $j:B\hookrightarrow\mathbb P^1$ omit these singular parameters. The endpoint action excludes global invariants of $L^-$. The parabolic Euler formula gives

$$ \dim H^1(\mathbb P^1,j_*L^-)=12+8+4+4-2\cdot4=20. \tag{B.13} $$

The three Eckardt lines are parametrized by

$$ T:\quad \zeta^2=-A(\beta)B(\beta),\qquad r=\frac{\zeta}{A(\beta)},\qquad A(\beta)=\prod_{i=1}^3(\beta-b_i),\quad B(\beta)=\prod_{i=4}^6(\beta-b_i). \tag{B.14} $$

This is the genus-two quotient of $G$; its degree-three map to the $r$-line has eight simple branch values. Its standard rank-two permutation summand is $L^+$. The intersection matrix of the three lines is $-2I+J$, nonsingular on that summand.

The twenty-seven line classes span the second cohomology of a smooth cubic surface. Their intersection matrix is $D-I$, where $D$ has eigenvalues $10,-5,1$ with multiplicities $1,6,20$ [28]. Thus (B.4) projects to $L^-$, on which the intersection pairing is $-6I$. The closure of the universal line family in $\mathcal C_{27}\times X$ gives an actual cylinder correspondence. Over $B$, Leray identifies $H^3(f^{-1}B)$ with $H^1(B,R^2f_*\mathbb Q)$, since the smooth rational surface fibers have $H^1=H^3=0$. Restriction of the projective line-cover normalization to the cover over $B$ is injective on $H^1$, with image the parabolic cohomology. The fiberwise nondegenerate intersection matrices therefore show that the cylinders inject

$$ H^1(T)(-1)\hookrightarrow H^3(X)^+, \qquad e_{\mathrm{line}}H^1(\mathcal C_{27})(-1) \hookrightarrow H^3(X)^-. \tag{B.15} $$

Their dimensions are four and twenty. Equation (B.11) makes both isomorphisms. In particular, no assertion about absence of boundary cohomology is needed to fill these eigenspaces.

B.3. Comparison with the specified quotient invariant

The dimension count alone would not identify the second space in (B.15) with the particular $\mathcal W$ from §10.3. We supply that comparison directly. Write $E:z^2=h^3+h$ and $G:v^4=\prod_i(q-b_i)$, with normalization at infinity. Delete $h=0,\infty$ from $E$, and the branch points and points above infinity from $G$, obtaining $E^\circ,G^\circ$. Set

$$ U=E^\circ\times \{(c_1,c_2)\in(G^\circ)^2:q(c_1)\ne q(c_2)\}. \tag{B.16} $$

The order-thirty-two group $H$ of §10.2 acts freely, and $U/H$ is an open of $X$. Indeed its coordinates are

$$ \begin{aligned} S&=q_1+q_2,& P&=q_1q_2,\\ x&=(v_1v_2)^2h,& y&=(v_1v_2)^3z,\\ t&=y/x,& w&=2x-t^2. \end{aligned} \tag{B.17} $$

Here $x\ne0$ and $H=\prod_iL_i\ne0$, so the birational chart is regular and the sextic and its small resolution agree.

Restriction $\mathcal W\to H^3(U)$ is injective. To prove this, the complement of $U$ in $M=E\times G\times G$ is a union of product-fiber divisors and $E\times\Gamma_{\tau^j}$, $0\le j\le3$, where $\tau(q,v)=(q,iv)$. The pure weight-three localization kernel is generated by Gysin images of $H^1$ of the divisor normalizations. Product-fiber terms lie in non-triple Künneth summands, whose $H$-invariants vanish. On $E\times\Gamma_{\tau^j}$, the $H^1(G)$ term does likewise. The $H^1(E)$ term maps to $H^1(E)\otimes[\Gamma_{\tau^j}]$ and is killed by the group average: the element $(\epsilon^2,\tau,\tau)$ acts as $-1$ on $H^1(E)$ and preserves each graph class. Thus the algebraic average $e_H$ annihilates every generator of this localization kernel.

Let $\Gamma\subset M\times X$ be the closure of the quotient graph. The finite étale norm identity on (B.16), followed by the injectivity just proved, gives the exact equality

$$ e_H\Gamma^*\Gamma_*=32\operatorname{id} \quad\hbox{on }\mathcal W. \tag{B.18} $$

Elliptic inversion corresponds under (B.17) to $t\mapsto-t$ and acts as $-1$ on $\mathcal W$. Hence $\Gamma_*\mathcal W\subset H^3(X)^-$. Both have dimension twenty, and its inverse is the algebraic correspondence $e_H\Gamma^*/32$. Composing this inverse with (B.15) proves (B.3) and the asserted specified identification (10.2).

B.4. Equations for the twenty-four-line source

Choose an Eckardt line $Z=0$, $P=\beta S$ over (B.14). The planes through it are $P=\beta S+\mu Z$. Define $a,b,c$ by

$$ A(\beta)B_3-B(\beta)A_3 =Z\bigl(aS^2+bZS+cZ^2\bigr) \quad\hbox{on }P=\beta S+\mu Z. \tag{B.19} $$

The residual conic is $aS^2+bZS+cZ^2-\zeta t^2=0$. Its four finite degenerate fibers and their line components give the normalization of

$$ b^2-4ac=0,\qquad \zeta^2=-A(\beta)B(\beta),\qquad \eta^2=\zeta a(\beta,\mu),\qquad r=\zeta/A(\beta). \tag{B.20} $$

These equations mean compactification followed by normalization, so vanishing affine denominators do not discard points. The factor in the last quadratic equation is $\zeta a$, rather than $ra$, because (B.19) uses the cleared coefficient. The order-four action is $(\zeta,\eta)\mapsto(-\zeta,i\eta)$.

Projection from a line on a cubic surface gives five degenerate conics; see [29, §2.1.2] for the classical construction. The fifth, at $\mu=\infty$, consists of the other two Eckardt lines. Each is fixed as a line by $\iota$, so this component disappears only after applying the anti-involution projector. Every non-Eckardt line meets exactly one Eckardt line; hence (B.20) describes all twenty-four, while (B.14) describes the remaining three. Formula (B.4), applied to this normalized cover, is the explicit tenfold. The entire anti-Prym need not equal its incidence image.

The external incidence theory is classical, and the separate global comparison (B.18) is essential to this application. In particular, the twenty-four-branch uniformization of principally polarized sixfolds in [28] is not applied to this different fibration. Neither this explicit Gaussian model nor its small resolution constructs a normal deformation to the generic orthogonal source: its whole third cohomology has rank twenty-four, while the minimal generic spin module has rank thirty-two.

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