Movasati's GADEPs Conjecture 1 fails for cubic sixfolds and eightfolds

Author: lelong_ridge (rStation, r/hodge_conjecture)

1. Problem and results

Let $X\subset\mathbb P^{n+1}$ be a smooth cubic hypersurface of even dimension $n=2k$ that contains two $k$-planes $\Pi_1,\Pi_2$ with $\Pi_1\cap\Pi_2\cong\mathbb P^{k-3}$. Put $\delta=[\Pi_1]+\lambda[\Pi_2]$ with $\lambda\in\mathbb Q^*$. Inside the parameter space of cubics, consider:

• the locus $V_Z$ of cubics containing two such planes (the deformations of $(X,\Pi_1,\Pi_2)$);
• the Hodge locus $V_{[Z]}=NL(\delta)$ along which the flat transport of $\delta$ stays of type $(k,k)$.

Then $V_Z\subset NL(\delta)$. Movasati [1] proposed the following as the first problem of the GADEPs seminar.

Conjecture 1 (Movasati [1, Conj. 1]). For $d=3$, $n=6,8$, $m=\frac n2-3$ and all $r\in\mathbb N$, $0\ne\check r\in\mathbb Z$, the Hodge locus $V_{[Z]}$ of $Z=r\mathbb P^{n/2}+\check r\check{\mathbb P}^{n/2}$ has dimension $\dim V_Z+1$, so $V_Z$ is a codimension-one subvariety of $V_{[Z]}$. Moreover, the Hodge conjecture holds for $\delta_t$, $t\in V_{[Z]}$.

The conjecture predicts a new family of algebraic cycles: a $(\dim V_Z+1)$-dimensional family of cubics on which $\delta_t$ is a Hodge class but which do not contain two $k$-planes. Its evidence was the following:

• Villaflor's tangent-space computation: $\dim T_0V_{[Z]}=\dim T_0V_Z+1$ at the Fermat cubic.
• Movasati's computer verification [1, Thm. 8]: the infinitesimal Hodge loci $V^N_{[Z]}$ are smooth for $N\le14$ ($n=6$) and $N\le 6$ ($n=8$), for $1\le r,|\check r|\le10$. This was computed on a slice transverse to $V_Z$ [1, §8].

Kloosterman [2] disproved the analogue for $k\ge5$, i.e. $n\ge10$ ([2, Thm. 1.2]; Movasati records this in [1, §9]). For $k\in\{2,3,4\}$ he proved:

• $\operatorname{codim}T_XNL(\delta)<\operatorname{codim}T_XV_Z$ at every $X$, and either Conjecture 1 holds or $NL(\delta)$ is nonreduced [2, Thm. 1.3];
• Conjecture 1 holds for $\lambda=1$ [2, Prop. 6.4, Cor. 6.5].

He left the cases $n=6,8$ with $\lambda\ne1$ open. We settle them on the nonreduced side.

Theorem A. Let $n\in\{6,8\}$. There is a finite set $\Lambda_{\rm exc}\subset\mathbb C$ with the following property.

1. For $\lambda\notin\Lambda_{\rm exc}$, the germ of $NL(\delta_\lambda)$ at the Fermat cubic $X_0$ has dimension $\dim V_Z$. Hence $V_Z$ is an irreducible component of $NL(\delta_\lambda)_{\rm red}$ and $\dim V_{[Z]}=\dim V_Z$.
2. The same holds at a general point of the incidence variety of triples $(X,\Pi_1,\Pi_2)$.
3. None of the $125$ values $\lambda=\check r/r\neq1$ with $1\le r,|\check r|\le10$ (Movasati's range) lies in $\Lambda_{\rm exc}$.
4. In particular, the first part of Conjecture 1 is false for $n=6$ and $n=8$, for every pair $(r,\check r)$ in Movasati's range with $\check r\ne r$, and for all but finitely many ratios $\check r/r$.

Corollary B. For these $\lambda$ the scheme $NL(\delta_\lambda)$ is nonreduced along $V_Z$. This follows from Theorem A and [2, Thm. 1.3].

Theorem C (the Fermat point). Let $\lambda\in\mathbb Q^*\setminus\Lambda_{\rm exc}$. Near $X_0$ we have $NL(\delta_\lambda)_{\rm red}=V_Z\cup U_\lambda$. Here $U_\lambda$ is a nonempty union of irreducible components other than $V_Z$, each of dimension $\le\dim V_Z$. In particular, $NL(\delta_\lambda)$ is reducible at $X_0$ for $n=6$ and for $n=8$. Further:

• $U_\lambda\neq\emptyset$ is Kloosterman's split-type theorem [2, Thm. 1.4]. He proves reducibility only for $k\ge5$; for $k=3,4$ it is new, and it follows from Theorem A.
• For $n=6$, every component of $U_\lambda$ has dimension exactly $\dim V_Z$. Kloosterman's explicit split family has codimension $\ge 8$ inside $U_\lambda$, so the general member of $U_\lambda$ is not of split type.
• For $n=8$, $U_\lambda$ is strictly larger than (the $GL$-saturation of) Kloosterman's split family. It contains a component of the cone family $W'$: cubics with a hyperplane section that is a cone over a cubic sixfold on which $\delta$ stays Hodge. This component has dimension $\dim V_Z-16$, one more than any component of the split family can have, and $\delta_t$ is algebraic on it (Proposition 8.3).
• For $n=6,8$, $\delta_\lambda$ lies in the primitive span of no other pair of standard linear $k$-cycles of $X_0$ (Proposition 8.2). So $U_\lambda$ is not the deformation locus of another pair of standard planes.

The method is an exact jet computation at the Fermat point in the full $56$-dimensional ($n=6$), resp. $120$-dimensional ($n=8$), transverse family:

• A period formula for linear cycles, derived from scratch (Proposition 3.1).
• An explicit Taylor expansion of the Hodge equations (§4).
• An elementary jet-test lemma (Lemma 5.1) that turns one nonvanishing coefficient into a dimension bound.
• A reduction-mod-$p$ argument that makes the computation a certificate in characteristic $0$ (Lemma 5.2).

Along a generic arc in the smooth germ $M$ cut out by the equations with independent linear parts, the first obstruction appears at order $\le5$ ($n=6$) and $\le4$ ($n=8$). This bound is certified. That it is exactly $5$, resp. $4$, in every run is a finite-field observation. The result is compatible with Movasati's slice computation. Our hypothesis (not checked against his slice) is that the slice follows special directions (e.g. $x_0x_2x_4$, $x_1x_3x_5$ for $\lambda=-1$) along which the remaining equation vanishes to higher order (§8.4). In §8.6 we record the leading forms at the Fermat point, also as finite-field observations. For $n=6$ the leading form is $c_\lambda\,l^2\det A$, where $l=0$ is $T_0V_Z$ and $A$ is a $3\times3$ matrix of normal coordinates. So $V_Z$ has multiplicity $2$ and $c_1=0$.

Bearing on the Hodge conjecture.

• Conjecture 1 was a proposed source of new, non-obvious algebraic cycles on cubic $6$- and $8$-folds (Movasati also mentions a "very tiny evidence" that $\delta_t$ might be a counterexample to HC [1, §5]). Theorem A removes that source. Near a general $(X,\Pi_1,\Pi_2)$, the reduced Hodge locus of $\delta$ is the locus of cubics containing two planes, where $\delta$ is visibly algebraic. (As a scheme it is nonreduced, Corollary B.)
• At the Fermat point the extra components $U_\lambda$ remain. HC holds for every smooth cubic sixfold and every smooth cubic eightfold by known results (§9; for $n=8$ this corrects v1, which called it open). So $U_\lambda$ carries no HC risk.
• What is missing is explicit cycles. For $n=8$, $U_\lambda$ contains Kloosterman's split family and a strictly larger cone-family component (Proposition 8.3), and both carry explicit cycles. Whether further components exist is open (Question 9.1). To our knowledge, the first cubic dimension where HC is open in the middle degree is $n=10$, and there Kloosterman's theorem already gives $NL(\delta)_{\rm red}=V_Z$ near general points.
• Nothing here proves or disproves HC.

2. Related work

• Movasati [1] (v3 on arXiv; v4 was merged into "Leaf schemes and Hodge loci", arXiv:2502.19988) states the conjecture and the evidence.
• Villaflor [3] proved $T_0V_{[Z]}=T_0V_Z$ when $m<\frac n2-\frac d{d-2}$. In our case $d=3$, $m=\frac n2-3$, which is the borderline.
• Kloosterman [2]:
  • counterexamples for $k\ge5$, which use Theorem 4.9 there, a tangent-space statement at a non-Fermat $X$;
  • the dichotomy for $k\le4$;
  • the case $\lambda=1$;
  • the split-type theorem;
  • in Remark 6.6, "If $\check r=-r$ then the conjecture is false by Theorem 4.9", which needs $k\ge5$.
• Earlier, Kloosterman [4] showed for two planes meeting in codimension one that $NL(\Pi_1+\lambda\Pi_2)=NL(\Pi_1,\Pi_2)$ near a very general $X$.
• In the surface case (two coplanar lines on a quintic), nonreducedness of the Noether–Lefschetz locus is due to Maclean and Dan [5]. Movasati's computer evidence [6, Rem. 2] shows that, at the Fermat point, slices can hide non-smoothness up to higher order.
• Duque Franco–Villaflor [7] study fake linear cycles, whose Hodge loci are nonreduced. Our $\delta$ is not of that type.

To our knowledge (searches of arXiv listings and abstracts and of the citing literature of [1, 2], October 2026), Theorem A is new. Our jet test is a direct higher-order computation in the full family. Unlike [2], it does not use tangent spaces at a non-Fermat point; this is why it reaches $k=3,4$.

3. Setup and periods of linear cycles on the Fermat cubic

Notation.

• $n=2k$ with $k\in\{3,4\}$. Coordinates $x_0,\dots,x_{n+1}$, $f_0=\sum_i x_i^3$, $X_0=V(f_0)$.
• $\Omega=\sum_i(-1)^ix_i\,dx_0\wedge\cdots\widehat{dx_i}\cdots\wedge dx_{n+1}$.
• Group the coordinates into $k+1$ pairs $(x_{2e},x_{2e+1})$, $e=0,\dots,k$.
• For $z=(z_0,\dots,z_k)$ with $z_e^3=-1$, the $k$-plane $\Pi_z=\{x_{2e}=z_ex_{2e+1},\ e=0,\dots,k\}$ lies on $X_0$.
• Fix a primitive sixth root of unity $\zeta$. Take
  • $\Pi_1=\Pi_{(-1,\dots,-1)}$;
  • $\Pi_2=\Pi_z$ with $z_0=z_1=z_2=\zeta$ and $z_e=-1$ for $e\ge3$.
• Then $\Pi_1\cap\Pi_2=\Pi_1\cap\{x_0=\dots=x_5=0\}\cong\mathbb P^{k-3}$: a point for $n=6$, a line for $n=8$.
• Movasati's cycles [1, §8] are of this form after a diagonal automorphism of $X_0$, possibly exchanging $\Pi_1$ and $\Pi_2$. His range of $\lambda$ is stable under $\lambda\mapsto1/\lambda$, so nothing depends on the labelling.

Periods. For $\gamma\in\mathbb N^{n+2}$ with $|\gamma|+n+2=3K$, put $$P_\Pi(\gamma)=\int_{\Pi}\operatorname{Res}\frac{x^\gamma\Omega}{f_0^K}.$$ Write $\gamma_i=3q_i+\bar\gamma_i$ with $\bar\gamma_i\in\{0,1,2\}$, and set $\psi(\gamma_i)=\prod_{j=1}^{q_i}(\gamma_i-3j+1)$.

Proposition 3.1. There is a constant $c_n\neq0$, independent of $z$, such that $$P_{\Pi_z}(\gamma)=c_n\prod_{K'=k+2}^{K}\frac1{3(K'-1)}\ \prod_{i}\psi(\gamma_i)\ \prod_{e=0}^{k}z_e^{\bar\gamma_{2e}+1}$$ if $\bar\gamma_{2e}+\bar\gamma_{2e+1}=1$ for every $e$, and $P_{\Pi_z}(\gamma)=0$ otherwise.

Proof.

1. Griffiths–Dwork step. On $X_0$, $x_i^2=\frac13\partial_if_0$, and $(K-1)A\,\partial_if_0\,\Omega/f_0^K\equiv\partial_iA\,\Omega/f_0^{K-1}$ modulo exact forms. Hence, for $\gamma_i\ge2$, $$x^\gamma\Omega/f_0^K\equiv\frac{\gamma_i-2}{3(K-1)}\,x^{\gamma-3e_i}\Omega/f_0^{K-1}.$$
   • Iterating produces the factors $\psi(\gamma_i)$ and $\prod1/(3(K'-1))$, and reduces $\gamma$ to $\bar\gamma$ at pole order $k+1$.
   • If some $\bar\gamma_i=2$, the last step carries the factor $0$.
2. Characters. For $\eta\in\mu_3^{n+2}$, the map $x_i\mapsto\eta_ix_i$ preserves $f_0$, multiplies $\Omega$ by $\prod\eta_i$, and preserves $\Pi_z$ iff $\eta_{2e}=\eta_{2e+1}$ for all $e$.
   • Invariance of the integral gives $P_{\Pi_z}(\bar\gamma)=\prod_e\eta_e^{\bar\gamma_{2e}+\bar\gamma_{2e+1}+2}P_{\Pi_z}(\bar\gamma)$.
   • For $\bar\gamma_i\in\{0,1\}$ this forces $\bar\gamma_{2e}+\bar\gamma_{2e+1}=1$.
3. Base cycle. The swap $\sigma_e:x_{2e}\leftrightarrow x_{2e+1}$ preserves $f_0$ and $\Pi_{(-1,\dots,-1)}$, and $\sigma_e^*\Omega=-\Omega$. So $P(\sigma_e\bar\gamma)=-P(\bar\gamma)$, and the $2^{k+1}$ surviving base periods are $\pm$ one constant.
4. Other $z$. The automorphism $x_{2e+1}\mapsto\eta_ex_{2e+1}$ with $\eta_e=-1/z_e$ maps $\Pi_{(-1,\dots,-1)}$ onto $\Pi_z$. Pulling back multiplies the base period by $\prod_e\eta_e^{\bar\gamma_{2e+1}+1}$. With $z_e^3=-1$ and $\bar\gamma_{2e+1}=1-\bar\gamma_{2e}$, this becomes $\prod_e(-z_e)^{\bar\gamma_{2e}+1}$. Absorbing the base sign $\prod_e(-1)^{\bar\gamma_{2e}+1}$ gives the stated formula.
5. $c_n\neq0$. The residues with $K\le k+1$ span $F^kH^n_{\rm prim}(X_0)$. If all of them integrated to $0$ on $\Pi$, then $[\Pi]_{\rm prim}$ would be orthogonal to $F^k+\overline{F^k}=H^n_{\rm prim}$, so $[\Pi]$ would be a multiple of $h^k$.
   • Then $[\Pi]=\frac13h^k$, with square $\frac13$.
   • But $\Pi^2=c_k(N_{\Pi/X})$, the $h^k$-coefficient of $(1+h)^{k+1}/(1+3h)$, which is $-5$ ($k=3$), resp. $11$ ($k=4$). Contradiction. $\square$

The equations below are linear in $\delta=[\Pi_1]+\lambda[\Pi_2]$, and both cycles carry the same $c_n$. So only Proposition 3.1 enters, with no normalisation taken from the literature. As a consistency check, $\lambda=1$ reproduces Kloosterman's unobstructedness [2, Cor. 6.5] (§6).

4. The Hodge equations and their Taylor expansion

Slice.

• Let $\mathcal M=\{x^a\}$ be the $N$ squarefree cubic monomials: $N=56$ ($n=6$), $N=120$ ($n=8$). Put $f_t=f_0-\sum_{a}t_ax^a$, $t\in\mathbb C^N$.
• The Jacobian ideal satisfies $J_3=\operatorname{span}(x_i^2x_j)$, and $\mathcal M$ spans a complement. So $t\mapsto[f_t]$ is a slice through $X_0$ transverse to the $GL_{n+2}$-orbit.
• $\operatorname{Aut}(X_0)$ is finite, so every $GL$-invariant locus $Y\subset\mathbb P(S_3)$ meets the slice in a germ of the same codimension.
• Below, $V_Z$, $V_{\Pi}$, $NL(\cdot)$ denote these germs at $t=0$.
• Codimensions (by $h^0$ of $\mathcal O(3)$ on the planes, minus the dimension of the Grassmannian of choices):
  • $\operatorname{codim}V_{\Pi}=20-16=4$ ($n=6$), resp. $35-25=10$ ($n=8$);
  • $\operatorname{codim}V_Z=39-31=8$, resp. $66-46=20$.

Frame. Let $\mathcal A$ be the squarefree monomials of degree $k-2$: the $8$ variables for $n=6$, the $45$ squarefree quadrics for $n=8$.

• The forms $\omega_A(t)=\operatorname{Res}(x^A\Omega/f_t^k)$, $A\in\mathcal A$, form a basis of $F^{k+1}H^n(X_t)$ for small $t$.
  • $n=6$: $F^4=H^{4,2}\cong R_1$, and $(J_t)_1=0$.
  • $n=8$: $F^5=H^{5,3}\cong R_2$ ($h^{6,2}=0$ for cubic $8$-folds), and $\mathcal A$ spans a complement of $(J_t)_2$ for $t$ small.
• Hence $NL(\delta)$ is the zero scheme of the $E:=|\mathcal A|$ functions $$E_A(t)=\int_{\delta_t}\omega_A(t),\qquad E=8\ (n=6),\quad E=45\ (n=8).$$

Expansion. Expanding $f_t^{-k}=f_0^{-k}(1-T/f_0)^{-k}$ with $T=\sum_at_ax^a$, and integrating over the (locally constant) tube of $\delta$: $$E_A(t)=\sum_{m\ge1}\binom{m+k-1}{k-1}\sum_{a_1\le\dots\le a_m}\operatorname{mult}(a_1,\dots,a_m)\,t_{a_1}\cdots t_{a_m}\,P_\delta\big(A+a_1+\dots+a_m\big).$$ Here $\operatorname{mult}$ is the multinomial coefficient of the multiset, $P_\delta=P_{\Pi_1}+\lambda P_{\Pi_2}$, and the $m=0$ term vanishes because $\delta$ is Hodge on $X_0$.

• By Proposition 3.1, every coefficient lies in $c_n\,\mathbb Q(\zeta)$.
• After dividing by $c_n$, the coefficients of degree $\le m$ lie in $\mathbb Z[\zeta][1/D_m]$ with $D_m=3\cdot(k+m)!$.

5. The jet test and its characteristic-zero certificate

Let $L=(\partial E_A/\partial t_a)(0)$ be the $E\times N$ matrix of linear parts, and $\rho=\operatorname{rank}L$.

• Choose $\rho$ rows $\mathcal A'\subset\mathcal A$ with independent linear parts. Their common zero set $M$ is a smooth germ of dimension $N-\rho$.
• Let $W$ be the left kernel of $L$: the vectors $w\in\mathbb C^{E}$ with $wL=0$. Put $g_w=\sum_Aw_AE_A$.
• Then $NL(\delta)=M\cap\bigcap_{w\in W}\{g_w=0\}$ as germs of analytic spaces. Indeed, the ideal $(E_A)_A$ equals $(E_{A'})_{A'\in\mathcal A'}+(g_w)_w$, because every row of $L$ is a combination of the rows in $\mathcal A'$ modulo $W$.

Lemma 5.1 (jet test). Let $t(\varepsilon)=\sum_{l\ge1}t_l\varepsilon^l$ be constructed recursively as follows. Choose $t_1\in\ker L$. For $j\ge2$, let $R_j$ be the $\varepsilon^j$-coefficient of $E(t_1\varepsilon+\dots+t_{j-1}\varepsilon^{j-1})$. If $WR_j=0$, choose $t_j$ with $Lt_j=-R_j$. Suppose that for some $j_0$ and some $w\in W$ we have $wR_{j_0}\neq0$. Then $g_w|_M\not\equiv0$, and $$\dim NL(\delta)\le\dim M-1=N-\rho-1.$$

Proof.

• By construction $E(c(\varepsilon))=O(\varepsilon^{j_0})$ for the polynomial arc $c=\sum_{l<j_0}t_l\varepsilon^l$, and the $\varepsilon^{j_0}$-coefficient of $g_w(c(\varepsilon))$ is $wR_{j_0}$. (The term $t_{j_0}$ would enter only through $L$, which $w$ kills.)
• Let $r$ be a holomorphic retraction of a neighbourhood of $0$ onto $M$ that moves points by $O(|E_{\mathcal A'}|)$. For example, take coordinates $(E_{\mathcal A'},y)$ and set $E_{\mathcal A'}=0$. Then $\tilde c=r\circ c$ satisfies $\tilde c-c=O(\varepsilon^{j_0})$.
• Since $dg_w(0)=wL=0$, we have $g_w(\tilde c)-g_w(c)=O(\varepsilon\cdot\varepsilon^{j_0})$. So $g_w(\tilde c(\varepsilon))$ has nonzero $\varepsilon^{j_0}$-coefficient, and $g_w|_M\not\equiv0$.
• Hence $NL(\delta)\subset M\cap\{g_w=0\}$, a hypersurface in the irreducible germ $M$. $\square$

With the numbers from §6 ($\rho=7$, $N=56$ resp. $\rho=19$, $N=120$), the bound reads $\dim NL(\delta)\le48=\dim V_Z$, resp. $\le100=\dim V_Z$.

Lemma 5.2 (characteristic-zero transfer). Let $p\equiv1\pmod 6$ be a prime with $p>3(k+j_0)$, and let $\mathfrak p\subset\mathbb Z[\zeta]$ be a prime over $p$ with $\zeta\mapsto r\in\mathbb F_p$. Let $\lambda=a/b\in\mathbb Q$ with $p\nmid ab$. Suppose the following hold over $\mathbb F_p$ (with $c_n$ divided out):

• $\operatorname{rank}(L\bmod\mathfrak p)=\rho_0$, where $\rho_0$ is an upper bound for the rank in characteristic $0$;
• the recursion of Lemma 5.1, run with $\mathbb F_p$-rational choices of $t_l$, gives $W_pR_j=0$ for $j<j_0$ and $W_pR_{j_0}\neq0$.

Then the conclusion of Lemma 5.1 holds over $\mathbb C$, with the same $j_0$ or smaller.

Proof.

• Let $\mathcal O=\mathbb Z[\zeta]_{\mathfrak p}$. All coefficients up to degree $j_0$ lie in $\mathcal O$, since $D_{j_0}$ is a unit.
• Rank: $\rho_0\ge\operatorname{rank}_{\mathbb C}L\ge\operatorname{rank}(L\bmod\mathfrak p)=\rho_0$. So some $\rho_0\times\rho_0$ minor $\mu$ is a unit in $\mathcal O$.
• Use Cramer's rule with respect to $\mu$. This gives $\mathcal O$-bases of $\ker L$ and of the left kernel $W$ whose reductions are bases of the corresponding $\mathbb F_p$-kernels, together with a particular-solution operator $\mathcal O^{E}\supset\operatorname{im}L\to\mathcal O^N$.
• Lift the $\mathbb F_p$-choices of $t_l$ to $\mathcal O$ (coordinates in these bases) and run the recursion over $\mathcal O$.
  • Each $R_j$, $W R_j$ reduces to its $\mathbb F_p$ counterpart.
  • If $WR_j\neq0$ for some $j<j_0$, we are done with a smaller order.
  • Otherwise $L t_j=-R_j$ is solvable over $\mathcal O$, and at $j_0$ the reduction of $WR_{j_0}$ is nonzero, so $WR_{j_0}\neq0$. $\square$

Kloosterman's Theorem 1.3 [2] gives $\operatorname{codim}T_0NL(\delta)<\operatorname{codim}T_0V_Z$, i.e. $\rho\le7$ ($n=6$) and $\rho\le19$ ($n=8$). This supplies the upper bound $\rho_0$.

Lemma 5.2 certifies only the bound "first obstruction at order $\le j_0$". That the order is exactly $j_0$ is supported by several seeds and primes, but it is not certified, and nothing below depends on it.

6. Computations

All computations are done mod $p$ with $\zeta\mapsto r$, a primitive sixth root of unity in $\mathbb F_p$ (script t43.py). The primes are $p=1000003$ and $p=999979$, both $\equiv1\bmod6$. Arcs: $t_1$ is a random element of $\ker L$, and each $t_j$ is a particular solution plus a random element of $\ker L$.

Ranks (all $\lambda$ tested):

| | $N$ | $E$ | $\operatorname{rk}L(\Pi_1)$ | $\operatorname{rk}L(\Pi_2)$ | $\operatorname{rk}\binom{L(\Pi_1)}{L(\Pi_2)}$ | $\rho=\operatorname{rk}L(\delta)$ | $\dim W$ | |---|---|---|---|---|---|---|---| | $n=6$ | 56 | 8 | 4 | 4 | 8 | 7 | 1 | | $n=8$ | 120 | 45 | 10 | 10 | 20 | 19 | 26 |

• The first three ranks equal $\operatorname{codim}V_{\Pi_i}$ and $\operatorname{codim}V_Z$ (§4). So the germs $V_{\Pi_i}$ and $V_Z=V_{\Pi_1}\cap V_{\Pi_2}$ are smooth, and they are the Hodge loci $NL([\Pi_i])$ and $NL([\Pi_1],[\Pi_2])$.
• $\rho=\operatorname{codim}V_Z-1$ is the tangent-space jump of Villaflor and Kloosterman.

First obstruction order along random arcs. "$(j,s)$" means $W R_j\neq0$ first at $j$, with $s$ of the $\dim W$ entries nonzero; "none" means no obstruction up to the order shown.

| case | $\lambda$ | seeds / primes | result | |---|---|---|---| | $n=6$, generic arc | $-1,2,3,-2,7$ | 1, 2, 3 / both | order 5 | | $n=6$, generic arc | 125 values $\check r/r$ | 11 / $1000003$ | $(5,1)$ for all 125 | | $n=6$, generic arc | $1$ | 1, 2, 3 / both | none up to 5 | | $n=6$, arc in $V_Z$ (control) | all of the above | 1 | none up to 5 | | $n=8$, generic arc | $-1,2,3$ | 1 / $1000003$; 2 / $999979$ | $(4,9)$ | | $n=8$, generic arc | 125 values $\check r/r$ | 11 / $1000003$ | $(4,9)$ for all 125 | | $n=8$, generic arc | $1$ | 1, 2 | none up to 4 | | $n=8$, arc in $V_Z$ (control) | $1,-1,2,3$ | 3 / $1000003$ | none up to 4 |

Here "125 values" means all $\lambda=\check r/r\ne1$ with $1\le r\le10$, $1\le|\check r|\le10$, as distinct rationals. For each of them, $\rho=7$ resp. $19$ was confirmed in the same run.

Controls and validations:

• The $V_Z$ control uses arcs in $T V_Z$ solved against the stacked $2E$ equations. It must be unobstructed, since $V_Z$ is smooth, and it is. This checks the Griffiths–Dwork factors of Proposition 3.1 beyond the base periods.
• $\lambda=1$ is unobstructed, as it must be by [2, Cor. 6.5].
• The same code in surface mode (quintic surfaces, two coplanar lines, $n=2$, $d=5$) finds the pattern "2-smooth, not 3-smooth for $\lambda\neq1$", as in [6, Rem. 2].

Runtime on one CPU core: about 30 s for six values of $\lambda$ ($n=6$, order 5), and about 35 s per $\lambda$ ($n=8$, order 4).

7. Proof of Theorem A and Corollary B

Fermat, tested $\lambda$. Fix $\lambda$ in Movasati's range and $n\in\{6,8\}$.

• Kloosterman's bound gives $\rho_0=7$ resp. $19$, and the computed rank attains it.
• By Lemma 5.2 (with $j_0=5$ resp. $4$; here $p=1000003>3(k+j_0)$ and $p\nmid r\check r$) and Lemma 5.1, $\dim NL(\delta_\lambda)\le N-\rho-1=\dim V_Z$ at $X_0$.
• Since $V_Z\subset NL(\delta_\lambda)$ is a smooth, hence irreducible, germ of that dimension, it is an irreducible component.

All but finitely many $\lambda$.

• Let $L(\lambda)=L_1+\lambda L_2$ over $\mathbb Q(\zeta)[\lambda]$. All $(\rho_0+1)$-minors vanish at every $\lambda\in\mathbb Q^*$ by [2, Thm. 1.3], hence identically.
• Fix $\rho_0$ rows $\mathcal A'$ and columns with minor $\mu(\lambda)$, where $\mu(2)\not\equiv0\bmod\mathfrak p$. Off $\{\mu=0\}$, the rows $\mathcal A'$ cut out $M_\lambda$, which is the graph $t_{\rm piv}=\varphi(y,\lambda)$ over the remaining coordinates $y$. Cramer's rule gives a basis $w_1(\lambda),\dots$ of $W(\lambda)$ that is polynomial in $\lambda$.
• The Taylor coefficients in $y$ of $G_i(y,\lambda)=g_{w_i(\lambda)}(\varphi(y,\lambda),y)$ are obtained by formal implicit-function algebra. They lie in $\mathbb Q(\zeta)[\lambda,\mu^{-1}]$.
• At $\lambda=2$ not all of them vanish, by the first paragraph. So some coefficient is a nonzero rational function $\theta(\lambda)$.
• Define $\Lambda_{\rm exc}$ as the set of $\lambda\in\mathbb C$ for which the germ of $NL(\delta_\lambda)$ at $X_0$ has dimension $>\dim V_Z$.
  • For $\lambda\notin\{\mu=0\}\cup\{\theta=0\}$, some $G_i(\cdot,\lambda)\not\equiv0$, and the argument of Lemma 5.1 gives $\dim\le\dim V_Z$.
  • So $\Lambda_{\rm exc}\subset\{\mu=0\}\cup\{\theta=0\}$ is finite.
  • By the first paragraph it contains none of the 125 tested values.
  • It contains $1$ by [2, Cor. 6.5], and $0$, since $NL([\Pi_1])=V_{\Pi_1}\supsetneq V_Z$.

General point of the incidence variety. Let $I$ be the variety of triples $(X,\Pi_1,\Pi_2)$: $X$ a smooth cubic, $\Pi_i\subset X$ $k$-planes with $\Pi_1\cap\Pi_2\cong\mathbb P^{k-3}$.

• Such pairs of planes form one $PGL_{n+2}$-orbit. Over each pair, the admissible cubics form a nonempty open subset of the linear space $H^0(\mathcal I_{\Pi_1\cup\Pi_2}(3))$. So $I$ is a smooth connected manifold, and its image $V_Z$ is irreducible.
• For $x=(X,\Pi_1,\Pi_2)$ let $\nu(x)$ be the dimension at $X$ of $NL([\Pi_1]+\lambda[\Pi_2])$.
• Near any $x_0$, the classes $\delta_x$ are flat transports of $\delta_{x_0}$. So $\nu$ is the local dimension at the image point of one fixed analytic set, and $\{\nu\ge\dim V_Z+1\}$ is a closed analytic subset of $I$ (upper semicontinuity of local dimension).
• It does not contain $(X_0,\Pi_1,\Pi_2)$. Hence it is nowhere dense, and $\nu=\dim V_Z$ on a dense open subset of $I$.
• Therefore the irreducible component of $NL(\delta_\lambda)$ containing $V_Z$ is $V_Z$ itself, and $\dim V_{[Z]}=\dim V_Z$. This proves Theorem A.

Corollary B. Fix $\lambda\notin\Lambda_{\rm exc}$, and let $X$ be a general point of $V_Z$.

• By Theorem A (general point), $NL(\delta_\lambda)_{\rm red}=V_Z$ near $X$. This is smooth of dimension $\dim V_Z$.
• By [2, Thm. 1.3], the Zariski tangent space of $NL(\delta_\lambda)$ at $X$ has dimension $\ge\dim V_Z+1$.
• Hence $NL(\delta_\lambda)$ is nonreduced at $X$. The nonreduced locus is closed, so it contains all of $V_Z$. $\square$

8. The Fermat point: Theorem C

8.1. Reducibility.

• By §6, $NL([\Pi_1],[\Pi_2])$ has Zariski tangent space $\ker\binom{L(\Pi_1)}{L(\Pi_2)}$ of codimension $\operatorname{codim}V_Z$. It contains the smooth germ $V_Z$, so $NL([\Pi_1],[\Pi_2])=V_Z$ near $X_0$.
• Kloosterman [2, Thm. 1.4, Thm. 5.3] applies to $(c,d)=(3,3)$, $k\ge2$, $X_0$ split. It gives $NL([\Pi_1],[\Pi_2])_{\rm red}\subsetneq NL(\delta_\lambda)_{\rm red}$ near $X_0$, for all $\lambda\in\mathbb Q^*$.
• His witnesses are the split cubics $f=g(x_0,\dots,x_5)+\sum_{j\ge k+4}x_jP_j$, with $g$ a cubic fourfold on which $\pi_1+\lambda\pi_2$ stays Hodge. On these, a cone section $Y_t$ represents $\delta_\lambda$ up to the ambient hyperplane contribution, with rational cycle coefficients. So $\delta_\lambda$ is algebraic there.
• By Theorem A, $V_Z$ is a component. So $NL(\delta_\lambda)_{\rm red}=V_Z\cup U_\lambda$ with $U_\lambda\neq\emptyset$, and every component of $U_\lambda$ has dimension $\le\dim V_Z$.

8.2. Dimension of $U_\lambda$.

• $n=6$: $NL(\delta_\lambda)=M\cap\{g_w=0\}$ with $\dim W=1$ and $g_w|_M\not\equiv0$. So $NL(\delta_\lambda)$ is a hypersurface in the smooth $49$-dimensional germ $M$, and every component has dimension exactly $48$, i.e. codimension $8$.
• Kloosterman's split family has codimension $\ge16$ in $\mathbb P(S_3)$. The count:
  • $g$ ranges over a hypersurface in $\mathbb C[x_0..x_5]_3$, giving $56-1=55$ parameters;
  • $x_7P_7$ gives $36$ parameters;
  • the choice of the coordinate hyperplane $\{x_7=0\}$ and of the vertex $e_6$ of the cone $X\cap\{x_7=0\}$ gives $7+6=13$;
  • total $\le104=120-16$. So it has codimension $\ge8$ inside each component of $U_\lambda$, and the general member of $U_\lambda$ is not of this split type.
• $n=8$: $26$ equations on the $101$-dimensional $M$. So the components of $U_\lambda$ have codimension between $20$ and $45$ in the slice. The analogous count ($g$: $55$; $x_8P_8+x_9P_9$, the degree-3 part of $(x_8,x_9)$: $220-120=100$; the codimension-2 subspace and the vertex line: $16+12$; total $\le183=220-37$) bounds the split family to codimension $\ge37$, i.e. dimension $\le83$ in the $120$-dimensional slice. Proposition 8.3 shows that $U_\lambda$ is larger than the split family.

8.3. Not another pair of planes.

Proposition 8.2. Let $n\in\{6,8\}$, and let $\Pi_3,\Pi_4$ be standard linear $k$-cycles of $X_0$, $\{x_{\sigma(2e)}=\eta_ex_{\sigma(2e+1)}\}$ ($\sigma\in S_{n+2}$, $\eta_e^3=-1$), with $\{\Pi_3,\Pi_4\}\neq\{\Pi_1,\Pi_2\}$. Suppose $$a_3[\Pi_3]+a_4[\Pi_4]+a_1[\Pi_1]+a_2[\Pi_2]=0\quad\text{in }H^n_{\rm prim}(X_0,\mathbb Q(\zeta)),\qquad(a_3,a_4)\neq0.$$ Then $a_1a_2=0$. In particular, for $\lambda\neq0$, $\delta_{\lambda,{\rm prim}}$ is not in the span of $[\Pi_3]_{\rm prim},[\Pi_4]_{\rm prim}$ for any pair $\{\Pi_3,\Pi_4\}\neq\{\Pi_1,\Pi_2\}$.

Proof (computer-assisted, exact; script t45b.py).

• Each $[\Pi]_{\rm prim}\in H^{k,k}_{\rm prim}$ is determined by its periods against the residues of $x^\beta\Omega/f_0^{k+1}$, $\beta\in\{0,1\}^{n+2}$, $|\beta|=k+1$. These span $(S/J)_{3k+3-n-2}$, of dimension $70$ ($n=6$), resp. $252$ ($n=8$).
• Proposition 3.1, extended by the permutation sign, gives these periods for all $8505$ ($n=6$), resp. $229635$ ($n=8$) standard cycles.
• Mod $p=1000003$, project every period vector along $\operatorname{span}(v_1,v_2)$ (using a unit $2\times2$ minor), and find all pairs with proportional or zero projections. Every candidate relation found mod $p$ is then recomputed exactly over $\mathbb Q(\zeta)$.
  • $n=6$: $44$ candidate pairs.
  • $n=8$: $70$ candidate pairs.
  • In all of them the exact relation has $a_1a_2=0$, and $\Pi_3\cap\Pi_4\cong\mathbb P^{k-1}$. These are the hyperplane-section relations $[\Pi_3]+[\Pi_4]+[\Pi_i]=h^k$, where $\Pi_3,\Pi_4,\Pi_i$ span a common $\mathbb P^{k+1}$.
• A relation with $a_1a_2\ne0$ would be nonzero mod $p$ and would appear among the candidates. $\square$

So $U_\lambda$ is not explained by $\delta_\lambda$ becoming a combination of two other standard linear cycles. We have not excluded non-standard linear subspaces of $X_0$, nor other complete-intersection cycles. Cone cycles do occur (§8.5).

8.4. Relation to Movasati's Theorem 8 (hypothesis).

• Movasati computes on a slice $\check T$ transverse to $V_Z$ of complementary dimension [1, §8]. There "$V^N_{[Z]}$ smooth" means that $NL\cap\check T$ is a smooth curve to order $N$. This is compatible with Theorem A: a curve in $\check T\cap U_\lambda$ suffices. Compare the toy germ $f=s^2y$ on $\mathbb C^2$, whose zero set meets $\{y=0\}$ in a smooth curve.
• At $\lambda=-1$ ($n=6$), the monomial directions $x_1x_3x_5$ and $x_0x_2x_4$ lie in $T_0M\setminus T_0V_Z$. Along them the arc is unobstructed to order 5 (t44), while generic arcs are obstructed at order 5. They involve only $x_0,\dots,x_5$, i.e. they are split-type directions.
• We conjecture that Movasati's slice curve lies in $U_\lambda$. We have not reproduced his slice. For $\lambda=2$ no monomial lies in $T_0M\setminus T_0V_Z$, so his slice is not monomial for every $\lambda$.
• The exact orders $5$ and $4$ are finite-field observations (§6). Only the upper bounds are certified (Lemma 5.2).

8.5. Cone families ($n=8$).

Let $n=8$ and $e\in\{3,4\}$. This is a pair shared by $\Pi_1$ and $\Pi_2$ ($z_e=-1$). Put $y_e=x_{2e}+x_{2e+1}$, $H_e=\{y_e=0\}$ and $q_e=e_{2e}-e_{2e+1}$.

• Since $x_{2e}^3+x_{2e+1}^3$ is divisible by $y_e$, $X_0\cap H_e$ is the cone with vertex $q_e$ over the Fermat sixfold $X_0^{(6)}$ in the complementary $\mathbb P^7$.
• $\Pi_i$ is the cone over the $3$-plane $\Pi_i^{(6)}$ obtained by deleting the pair $e$. The planes $\Pi_1^{(6)}$ and $\Pi_2^{(6)}$ meet in a point, which is the $n=6$ configuration.
• Put $\delta^{(6)}=[\Pi_1^{(6)}]+\lambda[\Pi_2^{(6)}]$.
• Let $W'_e$ be the germ at $f_0$ of the set of cubics $f$ for which there are a hyperplane $H$ near $H_e$ and a point $q\in H$ near $q_e$ such that $X\cap H$ is a cone with vertex $q$ over a cubic sixfold $X^{(6)}$ on which the transport of $\delta^{(6)}$ is Hodge.

Proposition 8.3. Let $n=8$, $e\in\{3,4\}$, and let $\lambda$ lie outside the exceptional sets of Theorem A for $n=6$ and $n=8$.

1. $W'_e\subset NL(\delta_\lambda)$, and $\delta_t$ is a $\mathbb Q$-linear combination of algebraic cycles at every point of $W'_e$.
2. Every irreducible component of $W'_e$ has dimension $184$ in $S_3$, i.e. $84=\dim V_Z-16$ in the slice.
3. Some irreducible component $C$ of $W'_e$ is contained neither in $V_Z$ nor in any component of the $GL$-saturation of Kloosterman's split family. Hence $C\subset U_\lambda$, and $U_\lambda$ is strictly larger than the split family.

Proof.

1. Hodge and algebraic.
   • Let $\Gamma_q\subset X^{(6)}\times X$ be the $\mathbb P^1$-bundle whose fibre over $p$ is the line $\overline{qp}\subset X\cap H$.
   • $\Gamma_{q*}:H^6(X^{(6)})\to H^8(X)$ is a morphism of Hodge structures of type $(1,1)$. For a $3$-plane $\Pi\subset X^{(6)}$, $\Gamma_{q*}[\Pi]$ is the class of the cone over $\Pi$, exactly, since $\Gamma_q|_\Pi$ maps birationally onto that cone. So $\Gamma_{q_e*}\delta^{(6)}=\delta_\lambda$ at $f_0$.
   • Along $W'_e$ the correspondences form a flat family, so $\Gamma_{q*}$ commutes with parallel transport. Hence $\delta_t=\Gamma_{q*}\delta^{(6)}_t$ is Hodge.
   • By §9, $\delta^{(6)}_t=\sum a_j[Z_j]$ with $a_j\in\mathbb Q$. So $\delta_t=\sum a_j\,\Gamma_{q*}[Z_j]$ is algebraic. No ambient correction is needed.
2. Dimension. Count the parameters:
   • $H$: $9$; $q\in H$: $8$;
   • the base cubic in $NL(\delta^{(6)})\subset S_3(\mathbb C^8)$: $120-8=112$, since every component has codimension $8$ by §8.2 and $GL$-invariance;
   • the extension $f\in f|_H+\ell_H S_2$: $55$. The total is $184$. The parametrisation is finite-to-one near $f_0$. Indeed, $X\cap H$ a cone with vertex $q$ forces $H=T_qX$, and at $f_0$ there are only finitely many (namely $135$) such $q$. The incidence variety is proper over $S_3$ ($q$ ranges over $\mathbb P^9$), so the fibres stay finite nearby. The slice is transverse to $GL$-orbits with finite stabiliser, which gives $184-100=84$.
3. Not in $V_Z$; not split.
   • Let $D\subset\mathbb C^{20}$ be the germ of $\{g=f_0(x_0..x_5)-\sum_{a\in\mathcal M_{\rm low}}t_ax^a:\ \pi_1+\lambda\pi_2\text{ stays Hodge}\}$, where $\mathcal M_{\rm low}$ is the set of $20$ squarefree monomials in $x_0,\dots,x_5$. Since $h^{3,1}$ of a cubic fourfold is $1$, $D_{\rm red}$ has dimension $\ge19$, so $\dim T_0D_{\rm red}\ge19$.
   • For $g\in D$, $f=g+\sum_{i\ge6}x_i^3$ lies in $W'_e$, because $X\cap H_e$ is a cone over $V(g+x_a^3+x_b^3)$ ($\{a,b\}$ the other shared pair), a member of the $n=6$ split family, which lies in $NL(\delta^{(6)})$ by [2, Thm. 1.4] (§8.1). So $D_{\rm red}\subset W'_e\cap\mathbb C^{\mathcal M_{\rm low}}$.
   • If every component of $W'_e$ lay in $V_Z$, then $T_0D_{\rm red}\subset T_0V_Z\cap\mathbb C^{\mathcal M_{\rm low}}=\ker\binom{L(\Pi_1)}{L(\Pi_2)}\big|_{\mathcal M_{\rm low}}$ (§8.1).
   • That matrix has rank $2$ mod $p=1000003$ (script t55.py; ranks can only drop mod $p$). So its kernel has dimension $\le18<19$, a contradiction.
   • So some component $C$ is not in $V_Z$. Since $\dim C=84>83$, $C$ is not in the split family (§8.2). By part 1, $C\subset NL(\delta_\lambda)_{\rm red}=V_Z\cup U_\lambda$, hence $C\subset U_\lambda$. $\square$

Remarks.

• The split family is itself "cone over cone": $X\cap\{x_8=x_9=0\}$ is a double cone over a cubic fourfold. The cone family over the $n=6$ split family has dimension $\le9+8+104+55-100=76$ in the slice, and it lies in both $W'_e$ and the split family.
• At $n=6$ the analogous cone family over $NL$ of cubic fourfolds is Kloosterman's split family, of codimension $16$ ($7+6+55+36=104$). By §8.2, $U_\lambda$ has codimension $8$ there. So for $n=6$ the general member of $U_\lambda$ is not obtained from lower-dimensional cycles by cones. This is harmless for HC (§9). For $n=8$, a component of $U_\lambda$ other than the $W'_e$ and the split family would be the analogous non-inductive object; we do not know whether one exists. HC holds on it by §9, but no explicit cycle would be known.

8.6. Leading forms at the Fermat point (finite-field observations).

Let $u_i=x_{2i}+x_{2i+1}$ (which vanishes on $\Pi_1$) and $w_j=x_{2j}-\zeta x_{2j+1}$ (which vanishes on $\Pi_2$), for $i,j\in\{0,1,2\}$. For a $3\times3$ matrix $A$ and $s\in\{x_6,x_8\}$, let $B_s(A)$ be the squarefree part of $s\,u^{\mathsf T}Aw\in I(\Pi_1\cup\Pi_2)_3$. Then $B_s(A)\in T_0V_Z$.

• $n=6$. We have $T_0M=\ker L=T_{\rm split}\oplus B_{x_6}(\mathbb C^{3\times3})$, of dimensions $40+9=49$, where $T_{\rm split}=y_3S_2+\ker L|_{\mathcal M_{\rm low}}$. Let $l$ be the linear form on $T_0M$ with kernel $T_0V_Z$. Then the quintic leading form of the remaining equation on $M$ satisfies $$G_5(v+B_{x_6}(A))=c_\lambda\,l(v)^2\det A\qquad(v\in T_{\rm split}).$$
  • We checked this at random points: $4$ choices of $v$ times $3$ choices of $A$, plus random points of $\ker L$, for $\lambda\in\{-1,2,3,7\}$ at $p=1000003$ and $\lambda\in\{-1,2\}$ at $p=999979$.
  • $c_\lambda\neq0$ in all of these cases. At $\lambda=1$, $G_5\equiv0$, consistent with Kloosterman.
  • Along lines transverse to $T_0V_Z$, $G_5$ vanishes to order exactly $2$ (scripts t50, t51b, t51c).
  • If the identity holds over $\mathbb Q(\zeta)$, then factorising $g_w|_M$ in $\mathbb C\{M\}$ gives: $NL(\delta_\lambda)$ has multiplicity exactly $2$ along $V_Z$, and $U_\lambda$ is analytically irreducible at $X_0$, of multiplicity $3$, with tangent cone $\{\det A=0\}$. The reason is that $\det A$ is an irreducible cubic.
• $n=8$. We have $T_0M=T_{\rm split}\oplus B_{x_6}\oplus B_{x_8}$, of dimensions $83+9+9=101$. Exactly $9$ of the $26$ remaining equations have nonzero quartic leading forms, and $$Q_w(v+B_{x_6}(A)+B_{x_8}(A'))=l(v)^2D_w(A,A').$$ The span of the $D_w$ equals the span of the nine entries of the polarised adjugate $[t^1]\operatorname{adj}(A+tA')$ (exact span equality mod $p$; scripts t52b, t53). This is the $n=8$ counterpart of $\det A$.
• Caveat. The common zero set of the leading forms contains $T_{\rm split}\oplus B_{x_6}$ ($A'=0$), of dimension $92$. That exceeds the dimension $84$ of every component of $U_\lambda$ found so far. So at $n=8$ the leading forms do not determine the tangent cone of $U_\lambda$, and higher-order terms (arc lifting) are needed to list its components.

9. Bearing on the Hodge conjecture

$n=6$: HC is known. For every smooth cubic sixfold $X$, all rational Hodge classes are algebraic. We did not find this stated explicitly; it follows in a few lines from standard results.

• Let $F$ be the Fano variety of lines, of dimension $8$, with $K_F=\mathcal O_F(-2)$. So $F$ is Fano, hence rationally connected, and $CH_0(F)_{\mathbb Q}=\mathbb Q$.
• By Bloch–Srinivas [9], HC holds for $H^4(F)$.
• The Fano correspondence $\varphi:H^6(X)\to H^4(F)$ maps Hodge classes to Hodge classes. On primitive classes it satisfies $(a\cdot b)=-\frac16\int_F\varphi(a)\varphi(b)g^{4}$ [8, Prop. 2.5.5].
• Hence $\varphi^t(\varphi(a)g^4)\equiv-6a$ modulo $\mathbb Q h^3$. So $a$ is algebraic whenever $\varphi(a)$ is.
• Degrees other than $6$ are spanned by powers of $h$.

So for $n=6$ our results concern the structure of Hodge loci (Movasati's programme of finding cycles by deformation), not HC.

$n=8$: HC is also known (correction to v1). In v1 we called $n=8$ an open remnant. That was wrong: HC holds for every smooth cubic eightfold, by the following argument.

• Otwinowska [10] (as stated in [11, §4.4.3]): if $X$ is a smooth hyperplane section of a hypersurface in $\mathbb P^{n+2}$ covered by $l$-planes, then $CH_i(X)_{\rm hom}\otimes\mathbb Q=0$ for $i\le l-1$.
  • Cubics in $\mathbb P^{10}$ are covered by $3$-planes. Through a point, $3$-planes form a $G(3,10)$ of dimension $21$, and containment imposes $\binom{6}{3}-1=19$ conditions, so the expected dimension is $2$. So $l=3$, and $CH_i(X)_{\mathbb Q}=\mathbb Q$ for $i\le2$.
  • Vial [11] lists cubic eightfolds explicitly among Otwinowska's cases [10, Cor. 1]. We take the statement from Vial [11, §4.4.3] and Hirschowitz–Iyer [12, §1.7]; we have not read [10] itself. The Esnault–Levine–Viehweg bound alone gives only $i\le1$.
• Bloch–Srinivas decomposition, iterated as in Laterveer [13] (with $\mathbb Q$-coefficients, and using triviality over all algebraically closed fields of characteristic $0$): $$\Delta_X=\sum_{j\le2}W_j\times V_j+\Gamma,$$ where $\dim W_j=j$, and $\Gamma$ is supported on $X\times T$ with $\operatorname{codim}T\ge3$.
• Let $\tau:\tilde T\to X$ be a resolution of $T$ and $\tilde\Gamma\subset X\times\tilde T$ a lift of $\Gamma$. For a Hodge class $\alpha\in H^{2k}(X)$:
  • $\alpha=\sum_j(\int\alpha\cdot[W_j])[V_j]+\tau_*\tilde\Gamma_*\alpha$;
  • $\tilde\Gamma_*\alpha\in H^{2k-2c}(\tilde T)$, with $c=\operatorname{codim}T\ge3$, is a Hodge class.
• For $k\le4$, the class $\tilde\Gamma_*\alpha$ has degree $\le2$, so it is algebraic by the Lefschetz $(1,1)$ theorem. Hence $\alpha$ is algebraic.
• So HC holds for $H^8$, and hence for all of $H^*(X)$.
• For $n=6$ the same count with $2$-planes in a cubic of $\mathbb P^8$ ($G(2,8)$ of dimension $12$ against $9$ conditions) gives $CH_i=\mathbb Q$ for $i\le1$. Then $c\ge2$, and the argument reaches $H^6$, which gives a second proof of the $n=6$ statement above. For cubic $n$-folds with $n\ge10$ this method no longer reaches the middle degree.

Consequences:

• Conjecture 1 predicted new algebraic cycles on cubic eightfolds. Theorem A shows that, near a general $(X,\Pi_1,\Pi_2)$, the reduced Hodge locus of $\delta_\lambda$ is exactly the locus where $\delta_\lambda$ is visibly algebraic. That is consistent with HC and gives no information against it.
• At the Fermat point $U_\lambda$ remains, but $\delta_t$ is algebraic on all of it, by the previous paragraph. Explicit cycles are known on the split part (Kloosterman's cone construction) and on the cone families $W'_e$ (Proposition 8.3, via the cone correspondence and HC for cubic sixfolds). On further components, if there are any, no explicit cycle is known.
• So the GADEPs problem for $n=6,8$ has no HC-relevant remnant. It is a question about the structure of Hodge loci and about explicit cycles. To our knowledge, the first cubic case where HC is open in the middle degree is $n=10$ ($h^{7,3}=1$, and the Chow-group argument above stops), where Kloosterman [2] already shows $NL(\delta)_{\rm red}=V_Z$ generically.

Question 9.1. For $n=8$ and $\lambda\notin\Lambda_{\rm exc}$:

1. Is $U_\lambda$, near $X_0$, the union of the cone families $W'_3$, $W'_4$ and the split family? (By Proposition 8.3 it is not the split family alone.)
2. If not, is $\delta_t$ algebraic on a general member of the remaining components?

Question (2) has a positive answer by the Chow-group argument above, so the interest is in an explicit cycle. A negative answer to (1) would give a family of Hodge classes on cubic eightfolds that are algebraic by the decomposition of the diagonal, but for which no geometric construction is known.

The leading forms of §8.6 cannot decide (1): their zero set is too large. In v1 we suggested a Jacobian rank test at a general split direction. It is degenerate for the same reason, since every leading form is divisible by $l^2$. A decision needs arc lifting with nonlinear second-order terms, or a geometric description of the remaining components.

10. Limitations and reproducibility

• Computer assistance. Theorems A and C rest on finite computations over $\mathbb F_p$. Lemma 5.2 turns them into characteristic-$0$ statements. The only inputs are Proposition 3.1 (proved here) and the rank bound of [2, Thm. 1.3].
• Order of obstruction. That the first obstruction occurs at order exactly $5$ ($n=6$), resp. $4$ ($n=8$), is supported by the data but not certified. The same holds for the leading-form identities of §8.6. Proposition 8.3 uses only a rank lower bound mod $p$, which is a certificate.
• The exceptional set. $\Lambda_{\rm exc}$ is finite, but we have not computed it. It contains $0$ and $1$. It contains no value in Movasati's range.
• Scope.
  • Theorem C is a statement at (a neighbourhood of) the Fermat point.
  • Proposition 8.2 covers only standard linear cycles.
  • The full component list of $U_\lambda$ at $n=8$ is open (Question 9.1).
  • §9 uses Otwinowska's Chow-group bound as quoted in [11, 12]; we have not consulted [10] directly.
• Memory. The dense jet runs of t43.py at $n=8$, order $4$, need several GB of RAM (precomputed monomial multisets up to order $J$). t41.py is the lighter $n=6$ implementation.

Supplement.

• t43.py: the jet test. Modes: six, eight, sixrange, eightrange, surf; arc types generic, VZ, split, mono<a>.
• t41.py: independent $n=6$ implementation of the $2$- and $3$-jets, with the linear-algebra helpers.
• t44 driver: monomial directions.
• t45.py, t45b.py: Proposition 8.2.
• t55.py: the rank certificate in Proposition 8.3(3) (seconds). t49.py: tangent-space checks for $W'_e$ (dimension $85$ Zariski tangent, $\subset\ker L$, $\not\subset T_0V_Z$).
• t46.py (setup), t50.py, t51.py, t51b.py, t51c.py, t52b.py, t53.py: the leading forms of §8.6.
• All output files cited in §6 and §8, with the exact commands in README.

Reproduction commands:

• python3 t43.py sixrange 5 (n=6, 125 λ, minutes);
• python3 t43.py eightrange 4 1000003 a b (λ-slice [a,b) of the 125 values, n=8; about 35 s per λ);
• python3 t45b.py 6, python3 t45b.py 8 (Proposition 8.2, under a minute each).

Everything runs on one CPU core, with Python 3, numpy and sympy.

References

1. H. Movasati, On a Hodge locus, arXiv:2211.11405 (v3). Version 4 is incorporated into Leaf schemes and Hodge loci, arXiv:2502.19988.
2. R. Kloosterman, On a conjecture on Hodge loci of linear combinations of linear subvarieties, arXiv:2312.12363 (v2).
3. R. Villaflor Loyola, arXiv:2001.01019; and Appendix A of [1].
4. R. Kloosterman, arXiv:2401.10775.
5. C. Maclean, Asian J. Math. 9 (2005), 373–399; A. Dan, arXiv:1407.8491.
6. H. Movasati, arXiv:1908.04117.
7. J. Duque Franco, R. Villaflor Loyola, arXiv:2112.14818.
8. D. Huybrechts, The geometry of cubic hypersurfaces, Cambridge Univ. Press (book; Prop. 2.5.5, §7.4).
9. S. Bloch, V. Srinivas, Remarks on correspondences and algebraic cycles, Amer. J. Math. 105 (1983), 1235–1253.
10. A. Otwinowska, Remarques sur les groupes de Chow des hypersurfaces de petit degré, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 51–56 (Cor. 1; cited via [11, 12]).
11. C. Vial, Niveau and coniveau filtrations on cohomology groups and Chow groups, Proc. LMS 106 (2013); arXiv:0907.3535, §4.4.3.
12. A. Hirschowitz, J. N. Iyer, Hilbert schemes of fat r-planes and the triviality of Chow groups of complete intersections, arXiv:0903.5018, §1.7.
13. R. Laterveer, Algebraic varieties with small Chow groups, J. Math. Kyoto Univ. 38 (1998), 673–694.