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Algebraic cycles and Fano threefolds of genus 7

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Pith's one-line read For very general prime Fano threefolds of genus 7, an explicit cycle on Y×Y is Abel–Jacobi trivial but nonzero in the Chow group, so Y admits no multiplicative Chow–Künneth decomposition.

desk verdict Promising negative answer to Question 1.3 for genus 7, but the proof omits the crucial identification between the dual-curve cycle and Z_Y; deserves a serious referee but needs a computation. read the letter →

arxiv 2608.12950 v1 pith:B6F7ZCOC submitted 2026-08-13 math.AG

Robert Laterveer This is my paper · ORCID
classification math.AG MSC 14C1514C2514C30
keywords algebraiccyclesChowgroupsmultiplicativeChow–KünnethdecompositionFanothreefoldsgenus7FranchettapropertyspinortenfoldAbel–Jacobitrivial
topics The Hodge Conjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that a very general prime Fano threefold of genus 7 carries an explicit algebraic cycle on its self-product that is invisible to the Abel–Jacobi map yet nonzero in the Chow group. Because such a cycle exists, these varieties do not admit a multiplicative Chow–Künneth decomposition, the current concrete form of the splitting-property conjecture. The proof transfers a known nontrivial zero-cycle from the square of the dual curve to the square of the threefold, and shows the transfer forces the failure of the Franchetta property for Y×Y. The paper also proves that every Fano threefold does admit such a decomposition modulo algebraic equivalence, so the obstruction is finer than algebraic equivalence.

What carries the argument

The load-bearing object is the explicit cycle $Z_Y$, a combination of the diagonal class and pullbacks of powers of the polarization. The proof is carried by a chain of equivalences: the Franchetta property for the universal family of genus-7 Fano threefolds, computed via the spinor-tenfold model; a Chow-level isomorphism of motives $h(Y)\cong h(C)(-1)\oplus \mathbf{1}\oplus \mathbf{1}(-3)$ relating $Y$ to its dual curve $C$; and generically defined correspondences from homological projective duality that transfer the obstruction. The cycle $Z_C=\Delta_C\cdot p_1^*K_C - \tfrac{1}{12}K_C\times K_C$ on $C\times C$ is the known nontrivial input.

What would settle it

Compute an explicit rational equivalence in $A^4(Y\times Y)$ for $Z_Y$ on a single very general genus-7 Fano threefold, or prove that the universal family $Y\times_B Y$ satisfies the Franchetta property in codimension 4; either would directly contradict Theorem 3.1.

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Extended reading notes

Core claim

The central result is Theorem 3.1: for a very general prime Fano threefold $Y$ of genus 7 (a smooth Fano threefold whose Picard group is generated by the canonical divisor), with $H=-K_Y\in A^1(Y)$, the cycle $$Z_Y = \Delta_Y\cdot (p_1)^*H - \tfrac{1}{12}\big((p_1)^*H\cdot(p_2)^*$H^{3}$ + (p_1)^*$H^{2}$\cdot(p_2)^*$H^{2}$ + (p_1)^*$H^{3}$\cdot(p_2)^*H\big) \in $A^{4}$(Y\times Y)$$ is homologically trivial but nonzero. The argument assumes $Z_Y=0$ and derives that the universal family $Y\times_B Y$ has the Franchetta property; the Chow-motive isomorphism between $Y$ and its dual curve $C$ then pulls that property back to $C\times C$, contradicting the known nontriviality of the interesting zero-cycle on $C\times C$. The contradiction also shows $Z_Y$ is Abel–Jacobi trivial. It follows that $Y$ has no multiplicative Chow–Künneth decomposition.

Load-bearing premise

The entire obstruction transfers through the Chow-level isomorphism of motives between $Y$ and its dual curve $C$; that isomorphism is only sketched, and if the correspondence realizing it fails to be generically defined or to induce the claimed injection on Chow groups, the conclusion collapses.

Editorial extensions

If this is right

  • Very general prime Fano threefolds of genus 7 do not admit a multiplicative Chow–Künneth decomposition.
  • The universal family $Y\times_B Y$ fails the Franchetta property in codimension 4, although it holds through codimension 3.
  • The Chow-motive correspondence between a genus-7 Fano threefold and its dual curve cannot preserve tautological rings even modulo algebraic equivalence.
  • Every Fano threefold admits a multiplicative Chow–Künneth decomposition modulo algebraic equivalence, so the negative answer to the original question is a rational-equivalence effect.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same transfer strategy may produce explicit nontrivial cycles on other Fano varieties with a dual curve or homological projective duality partner; the paper does not address those cases.
  • The cycle $Z_Y$ can be viewed as a candidate generator of the codimension-4 Griffiths group of $Y\times Y$; testing whether it spans that group over a very general fiber would sharpen the statement.
  • The contrast between Theorem 3.1 and Proposition 4.1 suggests that the right formulation of the splitting-property question for Fano threefolds is modulo algebraic equivalence, where the answer becomes uniformly positive.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies algebraic cycles on very general prime Fano threefolds of genus 7, in the context of Beauville's splitting property conjecture and Shen--Vial's multiplicative Chow--Künneth (MCK) decompositions. The main theorem exhibits an explicit cycle Z_Y in A^4(Y×Y), defined using the diagonal and powers of the anticanonical class, and claims that Z_Y is Abel--Jacobi trivial but nonzero; consequently Y admits no MCK decomposition. The proof passes to the dual genus-7 curve C via a Chow-motive isomorphism h(Y) ≅ h(C)(-1) ⊕ 1 ⊕ 1(-3), then transfers the nontriviality of the Faber--Pandharipande cycle on C×C (Green--Griffiths) to Y×Y. The paper also proves a positive result: every Fano threefold admits an MCK decomposition modulo algebraic equivalence.

Significance. If the main theorem is correct, it gives a negative answer to Question 1.3 for the genus-7 case, complementing the affirmative answers for cubic threefolds, intersections of two quadrics, intersections of a quadric and a cubic, and prime Fano threefolds of genera 8 and 10. The paper also provides one of the few explicit Abel--Jacobi-trivial nonzero cycles on a variety of dimension six. The strategy is attractive and largely reductionist: it uses the classical Green--Griffiths theorem for curves, with the 1/12 coefficient forced by -K_Y^3 = 12. The paper is also careful about the Franchetta property for Y and Y^2, and the mod-algebraic-equivalence statement for all Fano threefolds is a useful complement. The main caveat is that the central Chow-motive isomorphism is presented as a sketch, and the relative version needed for the transfer maps is not fully demonstrated.

major comments (3)
  1. [§2.5, Proposition 2.6] Proposition 2.6 is load-bearing, but its proof is only a sketch. The first option (homological isomorphism plus Kimura finite-dimensionality) does not specify the Chow correspondence f whose class is an isomorphism in M_hom; one must explicitly construct f and its inverse and check that the relevant Kimura-finiteness hypothesis applies to the endomorphisms before upgrading from M_hom to M_rat. The sentence 'check that it is induced by a correspondence' is not a proof. The second option is a one-sentence appeal to Bloch--Srinivas. The third option depends on assertions from [17, Thm. 4.4.11(iii)] and [15]/[16] about the flop, the 14 lines and 14 trisecants, and the identification of the common blow-up Z; these are not proved or quoted precisely enough for the role they play. Since all later transfer maps (4) and hence the contradiction in Theorem 3.1 depend on this Chow-level isomorphism, a complete proof or a precise reference with the full statement is required.
  2. [§2.5, Lemma 2.19 and §3, Eq. (4)] The relative version of the Chow-motive isomorphism is not established at the level needed for the transfer maps (4). Lemma 2.19 asserts that the correspondences of Proposition 2.6 are generically defined over B and induce an isomorphism GDA^2_B(Y) ≅ GDA^1_B(C)⊕Q^2, but the passage from the HPD Fourier--Mukai kernel to a relative Chow correspondence is only sketched; it does not justify why the relative correspondence induces an isomorphism of Chow motives fiberwise, as opposed to an isomorphism modulo homological equivalence. In addition, the universal family C→B of dual curves is never formally defined, although the groups GDA^i_B(C×C) in (4) presuppose such a family. Please define C→B and give the relative correspondence explicitly, or provide a precise reference for the relative HPD statement.
  3. [§3, proof of Theorem 3.1] In the last paragraph of the proof, the statement that 'the above argument actually shows that the map (4) sends Z_C to a non-zero multiple of Z_Y' is asserted without computation or reference. This is the only step that yields the Abel--Jacobi triviality of Z_Y. One needs either an explicit computation of the image of the Green--Griffiths cycle under the relative correspondence, or a theorem asserting that an isomorphism of Chow motives of the form h(Y) ≅ h(C)(-1)⊕1⊕1(-3) preserves the Abel--Jacobi kernel. As written, the full statement of Theorem 3.1 ('Abel--Jacobi trivial but non-zero') is not justified, even though the non-zeroness may follow from the preceding contradiction argument.
minor comments (4)
  1. [Conventions] The conventions paragraph appears to contain a typo: it defines A^j(Y) as the Chow group of j-dimensional cycles, but then says A^j(Y) and A^{n-j}(Y) are used interchangeably, which is the convention for codimension j. The body uses A^j as codimension. Please correct this.
  2. [§2.5, proof of Proposition 2.18] The statement that A^j_hom(Y)=0 for j≠2 is confusing without the codimension convention; A^2_hom(Y) may be nontrivial. The argument only needs the vanishing for the generically defined cycles, so please restate it in the notation used in the paper.
  3. [§3, proof of Theorem 3.1] The transition from Z_Y=0 to the Franchetta property for Y^2 is compressed. Please spell out that the relation kills all products of Δ with positive powers of p_i^*H, so that GDA^*_B(Y×Y) reduces to the pullback subalgebra plus Q[Δ] as a vector space; this is what makes the injectivity into cohomology immediate.
  4. [§4.2, proof of Corollary 4.5] The sentence 'the injective map B^*(C^m)→B^*(Y^m) provided by Proposition 2.6' refers to a statement that is only given for m=1. Please state explicitly that the map for m>1 is obtained by taking tensor products of the correspondence from Proposition 2.6.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the nonvanishing of Z_Y is reduced to the independent Green–Griffiths nonvanishing of the Faber–Pandharipande cycle Z_C, and the 1/12 coefficient is forced by -K_Y^3 = 12, not fitted.

full rationale

The derivation is not circular. Theorem 3.1 proves Z_Y is non-zero by contradiction: if Z_Y vanished, Propositions 2.18 and 2.20 would give the Franchetta property for Y×Y; the Chow-motive isomorphism of Proposition 2.6 together with Lemma 2.19 would then transfer this property to C×C, forcing the Faber–Pandharipande cycle Z_C to vanish, contradicting the external Green–Griffiths theorem [13]. The coefficient 1/12 is not fitted: it is forced by the genus-7 identity 2g-2 = 12 = -K_Y^3, matching the normalization of Z_C. The main cited inputs ([13], [17], [41], [42], [8]) are published theorems with proofs or external classifications, not restatements of Theorem 3.1. Proposition 2.20 is proved from the known [8, Proposition 5.2] rather than from the target result. The proof of Proposition 2.6 is admittedly sketched and the assertion that the map (4) sends Z_C to a non-zero multiple of Z_Y is not computed in detail, but these are correctness risks or gaps, not instances of a claim being equivalent to its input by construction. The paper's many self-citations are to earlier published works with independent content and do not define the obstruction in terms of its own nonvanishing. No equation in the paper reduces to its own input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters: the 1/12 in Z_Y is forced by -K_Y^3 = 12 = 2g−2. No invented entities; Z_Y is a constructed cycle, not a postulated object. The argument rests on a network of established results, several due to the author, which are used as published theorems rather than fitted inputs.

assumptions (5)
  • standard math Green-Griffiths theorem: for a very general curve of genus ≥4, Z_C := Δ_C·p_1^*K_C − 1/(2g−2) K_C×K_C is non-zero in CH^2(C×C)_Q.
    External theorem [13], with an alternative proof in [61]; the paper's contradiction relies directly on this non-vanishing.
  • standard math Mukai's classification and duality: prime Fano threefolds of genus 7 are dimensionally transverse linear sections of the spinor tenfold, and the dual curve construction yields a general curve of genus 7.
    Used in Theorem 2.2 and Proposition 2.4 to set up the geometry of Y and its dual curve C.
  • standard math Kimura finite-dimensionality for Fano threefolds and uniqueness of the even/odd decomposition of finite-dimensional motives.
    Used in Proposition 2.6 to upgrade the homological motive isomorphism to a Chow motive isomorphism; relies on Vial [56] and Kimura [19].
  • domain assumption The generalized Franchetta computation for Y^2: GDA^*_B(Y×Y) = ⟨(p_i)^*K_Y, Δ_Y⟩.
    Proposition 2.20 in the paper, established via [8, Prop 5.2] and the triviality of A^*(Σ_+); this is the bridge that translates the vanishing of Z_Y into the Franchetta property.
  • domain assumption The very general assumption: the dual curve C of a very general genus 7 Fano threefold is a very general curve of genus 7.
    Needed for the Green-Griffiths theorem to apply to the Faber-Pandharipande cycle on C×C.

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Pith. "Pith review of Algebraic cycles and Fano threefolds of genus 7." pith.science (2026). https://pith.science/paper/B6F7ZCOC

@misc{pith2026260812950,
  author       = {Pith},
  title        = {Pith review of: Algebraic cycles and Fano threefolds of genus 7},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B6F7ZCOC}},
  note         = {Machine review of arXiv:2608.12950}
}
abstract

Let $Y$ be a very general prime Fano threefold of genus 7. We exhibit an explicit 2-cycle on $Y\times Y$ that is Abel-Jacobi trivial but non-torsion in the Chow group $A^4(Y\times Y)$. As a consequence, $Y$ does not admit a multiplicative Chow-K\"unneth decomposition, in the sense of Shen-Vial. We also show that any Fano threefold has a multiplicative Chow-K\"unneth decomposition modulo algebraic equivalence.

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Pith tools

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