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 $NL(\delta_\lambda)$ 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 $n=6,8$, $\delta_\lambda$ 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 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 $\le4$ ($n=8$); in all runs it is exactly $5$, resp. $4$. This is compatible with Movasati's slice computation. We expect his slice to follow 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).

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 $(X,\Pi_1,\Pi_2)$ there is no family of cubics with $\delta$ Hodge beyond those containing two planes, on which $\delta$ is visibly algebraic.
• At the Fermat point the extra components $U_\lambda$ remain. For $n=6$ the Hodge conjecture holds on all smooth cubic sixfolds by known results (§9), so $U_\lambda$ carries no HC risk; but we do not know explicit cycles on its generic member.
• For $n=8$, HC for $\delta_t$ on $U_\lambda$ is, to our knowledge, open, unless $U_\lambda$ coincides with Kloosterman's split family, which we cannot decide. This is the only HC-relevant remnant of the GADEPs problem. We record it as a question (§9).
• 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, $\delta_\lambda$ is the class of a cone section $Y_t$, hence algebraic.
• 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 $\ge37$. We do not know whether $U_\lambda$ 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 $X_0$, $\{x_{\sigma(2e)}=\eta_ex_{\sigma(2e+1)}\}$ ($\sigma\in S_{n+2}$, $\eta_e^3=-1$). 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 or cone cycles.

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 $\lambda$.

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.

$n=8$: open remnant.

• For cubic eightfolds, $h^{5,3}=45\neq0$. We know no proof of HC in $H^8$ for arbitrary smooth cubic eightfolds.
  • The same correspondence lands in $H^6$ of the $12$-dimensional $F$.
  • The known bounds ($CH_0$ and $CH_1$ of $X$ are $\mathbb Q$; see [8, §7.4]) do not reach the middle degree.
• Conjecture 1 predicted new algebraic cycles there. Theorem A shows that, near a general $(X,\Pi_1,\Pi_2)$, the 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.
• What remains is $U_\lambda$ at the Fermat point. On its split part, $\delta_t$ is algebraic by Kloosterman's cone construction.

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

1. Is $U_\lambda$, near $X_0$, the $GL$-saturation of Kloosterman's split family?
2. If not, is $\delta_t$ algebraic on a general member of $U_\lambda$?

A negative answer to (1) would give an explicit, computable family of Hodge classes on cubic eightfolds for which no cycle is known.

A tangent-cone computation could decide (1) in favourable cases. The 26 obstruction forms on $M$ have leading terms of degree $4$. If their Jacobian at a general split direction has rank $18$ (the codimension of the split family in $M$), then the branch is the split family. We have not done this computation.

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 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 identity of $U_\lambda$ is open.

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.
• All output files cited in §6, 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, Hodge loci 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.