# RCHM × Hodge Conjecture — Research Round 2
## Constraint Superposition, Cancellation Domains, and Witness-Carrier Transmutation

**Status:** Internal research note  
**Version:** Round 2 / v0.2  
**Date:** 2026-09-17  
**Methodological base:** Relational Constraint–Handoff Methodology (RCHM) v0.1  
**Previous round:** RCHM × Hodge Conjecture — Research Round 1

---

## 0. Round 2 Objective

Round 1 decomposed the Hodge problem into:

$$
\text{Anchor defect}
+
\text{Handoff defect}
+
\text{Rigid defect}
+
\text{higher-order formal defect}.
$$

Round 2 asks a more primitive question:

> When several constraints or witness classes are combined, what is the operator by which they are combined?

The crucial correction is that the following two operations are not equivalent:

$$
\text{constraint conjunction}
$$

and

$$
\text{constraint superposition}.
$$

This distinction is essential because superposed constraints may cancel.

As a consequence:

$$
\boxed{
\text{adding relational structure can either remove freedom or generate new legal freedom.}
}
$$

This is a major extension of RCHM v0.1.

---

# 1. Two Different Meanings of “Satisfy Two Constraints”

Let $V$ be a local deformation space and let

$$
\Phi_1:V\to Q,
\qquad
\Phi_2:V\to Q
$$

be first-order obstruction maps.

The individual legal domains are

$$
L_1=\ker\Phi_1,
\qquad
L_2=\ker\Phi_2.
$$

There are at least two distinct ways to combine the constraints.

## 1.1 Conjunctive composition

Require both constraints separately:

$$
\Phi_1(v)=0,
\qquad
\Phi_2(v)=0.
$$

Then

$$
L_{\wedge}
=
L_1\cap L_2.
$$

Equivalently,

$$
L_{\wedge}
=
\ker(\Phi_1\oplus\Phi_2).
$$

This is ordinary constraint accumulation.

It can only preserve or remove freedom.

---

## 1.2 Superposed composition

Choose nonzero coefficients $a,b$ and impose only

$$
a\Phi_1(v)+b\Phi_2(v)=0.
$$

Then

$$
L_{a,b}
=
\ker(a\Phi_1+b\Phi_2).
$$

Necessarily,

$$
L_{\wedge}
\subseteq
L_{a,b}.
$$

But equality need not hold.

Indeed, it is possible that

$$
\Phi_1(v)\neq0,
\qquad
\Phi_2(v)\neq0,
$$

while

$$
a\Phi_1(v)
=
-b\Phi_2(v).
$$

Then

$$
v\notin L_1,
\qquad
v\notin L_2,
$$

but

$$
v\in L_{a,b}.
$$

Thus:

$$
\boxed{
\text{two individually illegal deviations can form a legally admissible relation.}
}
$$

This is not intersection.

It is relational cancellation.

---

# 2. First-Order Cancellation Quotient

Define the first-order relational cancellation domain

$$
\boxed{
\mathcal C^{(1)}_{a,b}
=
\ker(a\Phi_1+b\Phi_2)
\big/
(\ker\Phi_1\cap\ker\Phi_2).
}
$$

Then

$$
\mathcal C^{(1)}_{a,b}=0
$$

means that the superposed condition produces no new infinitesimal legal directions.

Conversely,

$$
\mathcal C^{(1)}_{a,b}\neq0
$$

means there exist directions legal only because the two obstruction channels cancel.

Define the first-order relational gain

$$
\boxed{
g_1(a,b)
=
\dim\mathcal C^{(1)}_{a,b}.
}
$$

Thus:

$$
g_1(a,b)>0
$$

is a precise linear-algebraic signature of relation-generated freedom.

This is the RCHM interpretation of what recent Hodge-locus literature calls **excess tangent dimension**.

---

# 3. Hodge-Locus Realization

Let $X$ be a smooth hypersurface and let

$$
\gamma_1,\gamma_2
\in
H^{2k}(X,\mathbb Q)\cap H^{k,k}(X)
$$

be Hodge classes.

Write

$$
\operatorname{NL}(\gamma)
$$

for the local Hodge locus where $\gamma$ remains of Hodge type.

Requiring both classes separately gives

$$
\operatorname{NL}(\gamma_1,\gamma_2)
=
\operatorname{NL}(\gamma_1)
\cap
\operatorname{NL}(\gamma_2).
$$

For the combined class

$$
\gamma_\lambda
=
\gamma_1+\lambda\gamma_2,
\qquad
\lambda\in\mathbb Q^\times,
$$

one always has

$$
\boxed{
\operatorname{NL}(\gamma_1,\gamma_2)
\subseteq
\operatorname{NL}(\gamma_1+\lambda\gamma_2).
}
$$

The inclusion may be strict.

At tangent level:

$$
T_X\operatorname{NL}(\gamma_1,\gamma_2)
\subseteq
T_X\operatorname{NL}(\gamma_1+\lambda\gamma_2).
$$

Kloosterman formalizes strict inclusion as excess tangent dimension.

RCHM identifies the quotient

$$
\boxed{
\mathcal C^{(1)}_\lambda
=
T_X\operatorname{NL}(\gamma_1+\lambda\gamma_2)
\big/
T_X\operatorname{NL}(\gamma_1,\gamma_2)
}
$$

as a cancellation-generated legal domain.

---

# 4. Why the Hodge Condition Can Cancel

Infinitesimally, the condition that $\gamma$ remain of type $(k,k)$ is linear in the class $\gamma$.

If the first-order Hodge obstruction associated with $\gamma_i$ is

$$
\Phi_{\gamma_i}(v),
$$

then linearity gives

$$
\Phi_{\gamma_1+\lambda\gamma_2}(v)
=
\Phi_{\gamma_1}(v)
+
\lambda\Phi_{\gamma_2}(v).
$$

Hence a direction $v$ can satisfy

$$
\Phi_{\gamma_1}(v)\neq0,
$$

$$
\Phi_{\gamma_2}(v)\neq0,
$$

but

$$
\Phi_{\gamma_1}(v)
+
\lambda\Phi_{\gamma_2}(v)
=
0.
$$

Therefore:

$$
\boxed{
\text{Hodge legality of a relation}
\not\Rightarrow
\text{Hodge legality of each component}.
}
$$

This gives a concrete mathematical model for the RCHM statement:

$$
\boxed{
\text{relational legality can be emergent.}
}
$$

---

# 5. Algebraic Hypersurface Formulation

For a smooth degree-$d$ hypersurface

$$
X\subset\mathbb P^{2k+1},
$$

the primitive cohomology is described through the Jacobian ring.

For a Hodge class $\gamma$, current hypersurface methods associate an Artinian Gorenstein ideal

$$
I(\gamma)\subset S
=
\mathbb C[x_0,\ldots,x_{2k+1}].
$$

The degree-$d$ piece determines the tangent space of the Hodge locus under the usual hypotheses:

$$
T_X\operatorname{NL}(\gamma)
\leftrightarrow
I(\gamma)_d.
$$

For two independent classes, Kloosterman proves that the excess tangent quotient can be written algebraically in terms of a left kernel of a bilinear pairing.

Schematically,

$$
\boxed{
\frac{
T_X\operatorname{NL}(\gamma_1+\lambda\gamma_2)
}{
T_X\operatorname{NL}(\gamma_1,\gamma_2)
}
\simeq
\ker_L(\psi_1+\nu(\lambda)\psi_2).
}
$$

This is a concrete implementation of the RCHM cancellation quotient.

The abstract obstruction-cancellation picture is therefore not merely metaphorical.

It already exists in computable commutative algebra.

---

# 6. When Cancellation Disappears

For two complete-intersection cycles $Y_1,Y_2$, Kloosterman gives conditions under which generic nonzero linear combinations have no excess tangent dimension.

For two $k$-planes intersecting in codimension $c$, a key numerical threshold is

$$
(c-1)(d-2)>2.
$$

In this regime, the generic combination behaves rigidly enough that

$$
T_X\operatorname{NL}([\Pi_1]+\lambda[\Pi_2])
=
T_X\operatorname{NL}([\Pi_1],[\Pi_2])
$$

for the relevant generic situation.

Thus, in RCHM language:

$$
\boxed{
\text{geometric separation}
+
\text{degree}
\Rightarrow
\text{suppression of relational cancellation}.
}
$$

The threshold is therefore interpretable as a **cancellation-permission boundary**.

Below or at the boundary,

$$
(c-1)(d-2)\le2,
$$

special cancellation behavior becomes possible.

Important exceptional regimes include

$$
(d,c)\in
\{(4,2),(3,3),(3,2)\}.
$$

---

# 7. A Concrete Dimension Example

Consider

$$
d=4,
\qquad
k=3,
\qquad
c=2.
$$

Thus $X$ is a quartic sixfold in

$$
\mathbb P^7,
$$

containing two $3$-planes intersecting in a line.

For one fixed $3$-plane, the condition that a quartic contain it imposes

$$
\binom{k+d}{d}
=
\binom{7}{4}
=
35
$$

linear conditions on the quartic polynomial.

The Grassmannian of $3$-planes in $\mathbb P^7$ has dimension

$$
(k+1)^2
=
16.
$$

Hence the locus containing some $3$-plane has expected codimension

$$
35-16
=
19.
$$

For an ordered pair of $3$-planes intersecting in codimension $2$, the space of such pairs has dimension

$$
(k+1)^2-c^2+2c(k+1)
=
16-4+16
=
28.
$$

A fixed pair imposes

$$
2\binom{7}{4}
-
\binom{5}{4}
=
70-5
=
65
$$

conditions.

Therefore the locus containing such a pair has codimension

$$
65-28
=
37.
$$

This provides a concrete baseline:

$$
\operatorname{codim}
\operatorname{NL}([\Pi_1],[\Pi_2])
=
37
$$

in the smooth expected situation.

The combined class may have a larger legal deformation domain than this joint locus.

That difference is precisely the relational cancellation sector.

---

# 8. The Most Important Geometric Finding: Witness-Carrier Transmutation

The strongest Round 2 observation is not the tangent calculation.

It is that the **type of algebraic witness itself can change under legal deformation**.

Consider the quartic case

$$
d=4,
\qquad
c=2.
$$

Kloosterman constructs a flat family

$$
(X_t,Y_t)
$$

such that for $t\neq0$, $Y_t$ is a complete intersection of multidegree

$$
(2,2,1,\ldots,1),
$$

while at the special fiber the primitive class satisfies

$$
[Y_0]_{\mathrm{prim}}
=
[\Pi_1]_{\mathrm{prim}}
-
[\Pi_2]_{\mathrm{prim}}.
$$

Thus the relation

$$
[\Pi_1]-[\Pi_2]
$$

does not need to remain represented as a difference of two independently deforming planes.

It can hand off to a new geometric carrier:

$$
\boxed{
\text{two-plane relation}
\longrightarrow
\text{complete-intersection witness}.
}
$$

This motivates the RCHM concept:

$$
\boxed{
\textbf{Witness-Carrier Transmutation}.
}
$$

A legal relation may preserve its cohomological identity while changing the geometric type of its witness.

---

# 9. Cubic Example: Relation to a Grassmannian-Type Carrier

There is an analogous phenomenon for

$$
d=3,
\qquad
c=3.
$$

For two $k$-planes intersecting in dimension $k-3$, Kloosterman constructs a flat family whose general witness is a linear section of an iterated cone over the Plücker embedding

$$
G(2,5)\hookrightarrow\mathbb P^9.
$$

At the special fiber, the class becomes, up to the ambient hyperplane contribution,

$$
[\Pi_1]+[\Pi_2].
$$

Thus:

$$
\boxed{
\text{sum of linear witnesses}
\longrightarrow
\text{Grassmannian-type witness}.
}
$$

This is an explicit example of the bridge-object mechanism proposed in Round 1.

It is no longer hypothetical.

The bridge object exists geometrically.

---

# 10. Consequence for Witness Search

Round 1 implicitly risked using a witness space of the form

$$
M_\alpha
=
\text{one Hilbert component of one geometric type}.
$$

Round 2 shows that this can be too restrictive.

The Hodge conjecture only requires a rational algebraic cycle:

$$
Z
=
\sum_i q_i Z_i,
\qquad
q_i\in\mathbb Q.
$$

It does not require:

- one irreducible subvariety;
- one fixed Hilbert component;
- preservation of the original decomposition;
- preservation of the original degree;
- preservation of the original carrier type.

Therefore the correct witness domain must permit **carrier changes**.

Instead of a single witness space, use a filtered union:

$$
\mathcal W^p(X)
=
\bigcup_{\tau}
\mathcal W^p_\tau(X),
$$

where $\tau$ indexes witness types.

Examples of $\tau$ may encode:

- number of irreducible components;
- degrees;
- complete-intersection type;
- Hilbert polynomial;
- Chow stratum;
- special homogeneous geometry;
- bridge-object type.

A deformation handoff may therefore have the form

$$
\mathcal W_{\tau_0}
\longrightarrow
\mathcal W_{\tau_1}
$$

with

$$
\tau_0\neq\tau_1.
$$

---

# 11. Witness Complexity Filtration

Introduce a monotone complexity function

$$
\mathfrak c(Z)
$$

on algebraic-cycle representations.

For a Hodge class $\alpha$, define its minimal known witness complexity by

$$
\boxed{
\kappa_X(\alpha)
=
\inf
\left\{
\mathfrak c(Z):
cl(Z)=\alpha
\right\}.
}
$$

More generally define

$$
\mathcal A^p_{\le B}(X)
=
\left\{
\alpha:
\exists Z,\;
cl(Z)=\alpha,\;
\mathfrak c(Z)\le B
\right\}.
$$

Then

$$
\mathcal A^p_{\le B}(X)
\subseteq
\mathcal A^p_{\le B'}(X)
\qquad
(B\le B').
$$

The full algebraic witness domain is

$$
\mathcal A^p(X)
=
\bigcup_B
\mathcal A^p_{\le B}(X).
$$

Thus the Hodge conjecture becomes:

$$
\boxed{
\forall\alpha\in\mathcal H^p(X),
\quad
\kappa_X(\alpha)<\infty.
}
$$

This formulation does not solve the conjecture.

Its purpose is algorithmic: witness search can expand systematically rather than assuming one carrier type.

---

# 12. Fake Linear Cycles as a Warning Against Carrier Identification

For Fermat hypersurfaces of degrees

$$
d=3,4,6,
$$

Duque Franco and Villaflor construct infinitely many fake linear cycles.

Their associated Artinian Gorenstein ideals have the same Hilbert-function behavior as those associated with genuine linear cycles, and their Hodge loci have maximal first-order tangent dimension.

However, they are not genuine linear cycles.

The authors further use second-order IVHS information to detect nonreduced Hodge-locus behavior.

In the relevant Fermat cases the Hodge conjecture is known, so these Hodge classes are algebraic, indeed rational combinations of linear cycles.

Therefore:

$$
\boxed{
\text{first-order carrier signature}
\not\Rightarrow
\text{actual carrier identity}.
}
$$

This is a critical RCHM warning.

An apparent interface may be an **alias** of another witness structure.

---

# 13. Interface Aliasing

Define two witness types $\tau_1,\tau_2$ to be order-$r$ interface aliases at a point if their observable Hodge-deformation data agree through order $r$ but their geometric carriers differ.

Symbolically:

$$
J^r_{\mathrm{Hodge}}(\tau_1)
=
J^r_{\mathrm{Hodge}}(\tau_2),
$$

while

$$
\tau_1\neq\tau_2.
$$

Fake linear cycles demonstrate that order-$1$ aliasing is possible.

Thus:

$$
\boxed{
\text{period/tangent reconstruction requires a depth parameter}.
}
$$

A carrier should not be identified from first-order data alone.

---

# 14. Jet-Level Cancellation Defect

Let

$$
\mathcal O
$$

be the local ring of the ambient deformation space at $X$, with maximal ideal $\mathfrak m$.

Let

$$
I_{\mathrm{joint}}
$$

be the local ideal of

$$
\operatorname{NL}(\gamma_1,\gamma_2),
$$

and

$$
I_{\lambda}
$$

the local ideal of

$$
\operatorname{NL}(\gamma_1+\lambda\gamma_2).
$$

Since

$$
\operatorname{NL}(\gamma_1,\gamma_2)
\subseteq
\operatorname{NL}(\gamma_1+\lambda\gamma_2),
$$

we have

$$
I_\lambda
\subseteq
I_{\mathrm{joint}}.
$$

Define the truncated local algebras

$$
A^{(r)}_\lambda
=
\mathcal O/
(I_\lambda+\mathfrak m^{r+1}),
$$

and

$$
A^{(r)}_{\mathrm{joint}}
=
\mathcal O/
(I_{\mathrm{joint}}+\mathfrak m^{r+1}).
$$

Define the order-$r$ cancellation defect

$$
\boxed{
\delta_r(\lambda)
=
\dim_{\mathbb C}A^{(r)}_\lambda
-
\dim_{\mathbb C}A^{(r)}_{\mathrm{joint}}.
}
$$

Then

$$
\delta_r(\lambda)\ge0.
$$

Interpretation:

- $\delta_1>0$: first-order relational freedom;
- $\delta_1=0$ but $\delta_2>0$: hidden second-order difference;
- several initial $\delta_r=0$ followed by $\delta_{r+1}>0$: delayed relational branching;
- persistent equality at all formal orders suggests formal coincidence;
- reduced loci can still require separate global/component analysis.

This extends the Round 1 defect ladder into a comparison of **joint legality** and **relation legality**.

---

# 15. Constraint Algebra

Round 2 suggests that RCHM needs an explicit algebra of constraint composition.

Let

$$
\Gamma_1,\ldots,\Gamma_m
$$

be constraint channels.

At least four distinct composition operators should be separated.

## 15.1 Conjunction

$$
\Gamma_{\wedge}
=
\Gamma_1\wedge\cdots\wedge\Gamma_m.
$$

Effect:

$$
\text{freedom decreases}.
$$

---

## 15.2 Superposition

For obstruction-valued constraints:

$$
\Gamma_a
:
\sum_i a_i\Phi_i=0.
$$

Effect:

$$
\text{freedom may increase through cancellation}.
$$

---

## 15.3 Quotient

Identify states related by a symmetry or equivalence:

$$
D/\sim.
$$

Effect:

$$
\text{apparent freedom decreases by identification}.
$$

---

## 15.4 Handoff

A state preserving an invariant changes carrier:

$$
(W_{\tau_0},\alpha)
\longrightarrow
(W_{\tau_1},\alpha).
$$

Effect:

$$
\text{representation changes while relational identity survives}.
$$

Therefore one may not infer system behavior from the number of constraints alone.

One must specify the composition operator.

---

# 16. New RCHM Principle: Constraint Composition Non-Monotonicity

A naive principle says:

$$
\text{more constraints}
\Rightarrow
\text{less freedom}.
$$

Round 2 shows that this is true for conjunction but false for superposition.

The corrected principle is:

$$
\boxed{
\text{constraint count does not determine effective freedom;}
\text{ constraint composition does.}
}
$$

More formally:

$$
\boxed{
\ker(\Phi_1\oplus\Phi_2)
\subseteq
\ker(a\Phi_1+b\Phi_2),
}
$$

and strict inclusion occurs exactly when there is nontrivial cancellation.

This is the simplest rigorous model of relationally generated admissibility.

---

# 17. New RCHM Principle: Witness Identity Is Not Carrier Identity

The cohomological object and its geometric carrier must be separated.

Let

$$
\alpha
$$

be the transported invariant class.

A witness at time $t$ is some

$$
W_t
$$

such that

$$
cl(W_t)=\alpha_t.
$$

There is no requirement that

$$
W_t
$$

remain in one fixed geometric family.

Therefore:

$$
\boxed{
cl(W_{t_0})=cl(W_{t_1})
\text{ under transport}
}
$$

does not imply

$$
\boxed{
\operatorname{Type}(W_{t_0})
=
\operatorname{Type}(W_{t_1}).
}
$$

This distinction is necessary for any global witness-search strategy.

---

# 18. Revised Hodge Attack Architecture

Round 1 proposed:

$$
\text{legal state}
\to
\text{anchor witness}
\to
\text{deformation handoff}
\to
\text{formal lifting}.
$$

Round 2 revises this to:

$$
\boxed{
\text{legal state}
\to
\text{witness ensemble}
\to
\text{constraint composition}
\to
\text{cancellation analysis}
\to
\text{carrier handoff}
\to
\text{jet-depth verification}
\to
\text{global closure}.
}
$$

The witness ensemble must permit type changes.

The constraint layer must distinguish conjunction from superposition.

The jet layer must prevent false identification caused by tangent-space aliasing.

---

# 19. Practical Search Algorithm for the Next Round

Given a Hodge class represented initially as

$$
\gamma
=
\sum_{i=1}^r a_i[Z_i],
$$

perform:

### Step 1 — Individual obstruction extraction

Construct

$$
\Phi_i.
$$

### Step 2 — Joint legal domain

Compute

$$
L_{\wedge}
=
\bigcap_i\ker\Phi_i.
$$

### Step 3 — Superposed legal domain

Compute

$$
L_a
=
\ker\left(\sum_i a_i\Phi_i\right).
$$

### Step 4 — First-order relational gain

Compute

$$
\mathcal C^{(1)}_a
=
L_a/L_{\wedge}.
$$

### Step 5 — Search for geometric carrier replacement

For each new direction in $\mathcal C^{(1)}_a$, ask whether the combined class admits a witness of another type:

$$
\tau_0\to\tau_1.
$$

### Step 6 — Jet verification

Compute or approximate

$$
\delta_2,\delta_3,\ldots.
$$

### Step 7 — Carrier-complexity expansion

If no witness exists in current type budget $B$, enlarge

$$
\mathcal W_{\le B}
\to
\mathcal W_{\le B'}.
$$

### Step 8 — Global component closure

Determine whether local directions form:

- one reduced component;
- a nonreduced component;
- multiple branches;
- an actual larger Hodge locus.

---

# 20. Immediate Next Target

The strongest next test is no longer a single linear cycle.

It is a pair

$$
\Pi_1,\Pi_2
$$

with controlled intersection.

Two especially useful regimes are:

$$
(d,c)=(4,2)
$$

and

$$
(d,c)=(3,3).
$$

They contain all ingredients needed to test RCHM:

- explicit Jacobian-ring control;
- explicit Artinian Gorenstein ideals;
- excess tangent dimensions;
- special coefficient values $\lambda$;
- reducible Hodge loci;
- nonreduced behavior;
- carrier transmutation;
- explicit flat families;
- complete-intersection witnesses;
- Grassmannian-type witnesses.

Thus the next research round should attack the map

$$
\boxed{
\lambda
\longmapsto
\left(
g_1(\lambda),
\delta_2(\lambda),
\operatorname{CarrierType}(\lambda)
\right).
}
$$

The goal is to determine whether coefficient space itself has a finite stratification into distinct relational regimes.

This would turn the informal question

> “What happens when two legal infinities intersect?”

into a computable version:

$$
\boxed{
\text{Given two constraint channels and a coupling coefficient,
what legal domain and witness carrier are generated?}
}
$$

---

# 21. Round 2 Conclusions

Round 2 produces five main conclusions.

### Conclusion A — Intersection is not enough

$$
\boxed{
\text{joint legality}
\neq
\text{superposed legality}.
}
$$

### Conclusion B — Relations can generate freedom

$$
\boxed{
\mathcal C^{(1)}_{a,b}\neq0
}
$$

means freedom exists only because two illegal deviations cancel.

### Conclusion C — Constraints require an operator

$$
\boxed{
\text{constraint composition}
\text{ determines freedom more strongly than constraint count}.
}
$$

### Conclusion D — Witness carriers can change

$$
\boxed{
\text{cohomological identity can survive geometric carrier transmutation}.
}
$$

### Conclusion E — First-order signatures are insufficient

$$
\boxed{
\text{tangent agreement}
\not\Rightarrow
\text{carrier identity}
\not\Rightarrow
\text{scheme-level equality}.
}
$$

The resulting upgraded RCHM chain is:

$$
\boxed{
\text{Domain}
\to
\text{Constraint Operator}
\to
\text{Cancellation Sector}
\to
\text{Witness Ensemble}
\to
\text{Carrier Handoff}
\to
\text{Jet Verification}
\to
\text{Global Closure}.
}
$$

This is the main methodological output of Round 2.

---

# References

1. Remke Kloosterman, **On a conjecture on Hodge loci of linear combinations of linear subvarieties**, Rendiconti del Circolo Matematico di Palermo Series 2, 74, Article 187 (2025).  
   DOI: https://doi.org/10.1007/s12215-025-01307-4

2. Jorge Duque Franco and Roberto Villaflor Loyola, **On fake linear cycles inside Fermat varieties**, Algebra & Number Theory 17 (2023), 1847–1865.  
   arXiv: https://arxiv.org/abs/2112.14818  
   DOI: https://doi.org/10.2140/ant.2023.17.1847

3. Roberto Villaflor Loyola, **Small codimension components of the Hodge locus containing the Fermat variety**, Communications in Contemporary Mathematics 24 (2022).  
   arXiv: https://arxiv.org/abs/2001.01019

4. Remke Kloosterman, **Variational Hodge conjecture for complete intersections on hypersurfaces in projective space**, Rendiconti del Seminario Matematico della Università di Padova 148 (2022), 185–201.

5. Hossein Movasati, **Leaf schemes and Hodge loci**, preprint (2025).  
   arXiv: https://arxiv.org/abs/2502.19988

6. Ananyo Dan and Inder Kaur, **Semi-regular varieties and variational Hodge conjecture**, Comptes Rendus Mathématique 354 (2016), 297–300.  
   DOI: https://doi.org/10.1016/j.crma.2016.01.005