Strong F-conjecture in Moduli of Curves

• The Strong F-conjecture is a statement that every F-nef divisor on the moduli space M₀,n is Q-linearly equivalent to an effective combination of boundary divisors.
• It connects combinatorial boundary relations and linear inequalities, with proofs in cases like M₀,7 and computer-assisted results in M₀,8.
• Its verification propagates implications to higher genera (up to g = 44) while distinguishing between the boundary-effectivity and the broader nefness criteria.

Searching arXiv for the specified paper and closely related work on the Strong F-conjecture in moduli of curves. arXiv search query: (Fedorchuk et al., 16 Jul 2025) The Strong F-conjecture is a conjectural statement about the geometry of the Deligne–Mumford moduli space $\overline{M}_{0,n}$ of stable $n$-pointed rational curves. In its standard form, it asserts that every F-nef divisor on $\overline{M}_{0,n}$ is $\mathbb{Q}$–linearly equivalent to an effective combination of boundary divisors. It is closely related to the broader F-conjecture, which gives a conjectural description of the ample cone of $\overline{M}_{g,n}$, and recent work has connected the strong form to the equivalence of the symmetric and non-symmetric F-conjectures, proved the Strong F-conjecture for $\overline{M}_{0,8}$, and deduced the F-conjecture for $\overline{M}_g$ up to genus $44$ (Fedorchuk et al., 16 Jul 2025).

1. Moduli-theoretic setting

Let $\overline{M}_{g,n}$ be the Deligne–Mumford moduli space of stable $n$–pointed curves of genus $n$0. Its Picard group is generated over $n$1 by the boundary divisors $n$2 parameterizing nodal curves with markings split into two disjoint subsets $n$3, together with the cotangent-line classes $n$4.

A one-dimensional boundary stratum of $n$5 is called an F-curve. A divisor $n$6 is called F-nef if for every F-curve $n$7 one has $n$8.

The F-conjecture for $n$9 states that a divisor on $\overline{M}_{0,n}$0 is nef if and only if it is F-nef. In particular the nef cone is cut out by finitely many linear inequalities. The Strong F-conjecture specializes to genus $\overline{M}_{0,n}$1 and replaces the numerical nefness criterion by a boundary-effectivity statement: every F-nef divisor on $\overline{M}_{0,n}$2 is $\overline{M}_{0,n}$3–linearly equivalent to an effective combination of boundary divisors (Fedorchuk et al., 16 Jul 2025).

This formulation makes the strong form substantially more rigid than the bare nefness statement. A plausible implication is that the conjecture is best understood not only as a criterion for nefness, but also as a problem in the combinatorics of the boundary basis and the relations among boundary classes.

2. Boundary-basis formulation and F-curve inequalities

In boundary-basis notation one writes any divisor $\overline{M}_{0,n}$4 on $\overline{M}_{0,n}$5 as

$\overline{M}_{0,n}$6

The strong form demands $\overline{M}_{0,n}$7 after adding relations in $\overline{M}_{0,n}$8, namely the Keel relations (Fedorchuk et al., 16 Jul 2025).

An F-curve in $\overline{M}_{0,n}$9 corresponds to a partition

$\mathbb{Q}$0

into four nonempty parts. For such a curve,

$\mathbb{Q}$1

Requiring all these pairings to be nonnegative is the F-nef condition.

A second description writes divisors on $\mathbb{Q}$2 in terms of weights $\mathbb{Q}$3:

$\mathbb{Q}$4

Here $\mathbb{Q}$5 is equivalent to $\mathbb{Q}$6 being an effective boundary. For an F-curve $\mathbb{Q}$7 one has

$\mathbb{Q}$8

Checking F-nefness is therefore a finite list of linear inequalities in the $\mathbb{Q}$9.

This linear-algebraic reformulation is central to all presently known small-$\overline{M}_{g,n}$0 proofs. It suggests that the Strong F-conjecture is simultaneously a statement about divisor cones, linear inequalities, and effective redistributions of weights across the boundary basis.

3. Symmetric reduction and equivalence with the full F-conjecture

Let $\overline{M}_{g,n}$1 act by permuting the $\overline{M}_{g,n}$2 markings on $\overline{M}_{g,n}$3. One may formulate a weaker symmetric F-conjecture which only tests $\overline{M}_{g,n}$4-invariant divisors against all F-curves. Theorem (Fedorchuk–Mellit): the full non-symmetric F-conjecture for all $\overline{M}_{g,n}$5 is equivalent to the symmetric F-conjecture. In particular, to prove that every F-nef divisor on $\overline{M}_{g,n}$6 is nef it suffices to treat only $\overline{M}_{g,n}$7-invariant divisors (Fedorchuk et al., 16 Jul 2025).

The proof proceeds by transforming an arbitrary F-nef divisor into a pullback of an $\overline{M}_{g,n}$8–symmetric F-nef divisor on a larger moduli space. Given any not necessarily symmetric F-nef divisor $\overline{M}_{g,n}$9 on $\overline{M}_{0,8}$0, one constructs—via an attaching-map trick, the Chinese Remainder Theorem on labels, and a carefully chosen “supertotal” F-nef function on a large cyclic group—a pullback of an $\overline{M}_{0,8}$1–symmetric F-nef divisor on $\overline{M}_{0,8}$2. Concretely, one embeds the original labelling set $\overline{M}_{0,8}$3 into residue classes mod a large integer

$\overline{M}_{0,8}$4

so that the pullback of a symmetric function recovers $\overline{M}_{0,8}$5.

The significance of this equivalence is structural. It shows that any putative counterexample to the general F-conjecture would already violate the symmetric case. This suggests that the symmetric cone is not merely a simplification for computation, but a genuinely complete testing ground for the conjecture in genus $\overline{M}_{0,8}$6.

4. Verified cases: $\overline{M}_{0,8}$7 and $\overline{M}_{0,8}$8

A key combinatorial reduction shows that for $\overline{M}_{0,8}$9-invariant divisors it suffices to check effectivity on pulling back to boundary strata of type attaching-maps for strict integer partitions

$\overline{M}_g$0

with $\overline{M}_g$1. This uses an “ascent of effectivity” lemma: when two of the parts coincide one bundles them together, uses F-nef inequalities, and redistributes weights to produce a valid boundary-weight function (Fedorchuk et al., 16 Jul 2025).

For $\overline{M}_g$2, an alternative proof originally due to Larsen checks by hand that the finitely many $\overline{M}_g$3–invariant F-nef divisors are stratally effective. All possible strict partitions of $\overline{M}_g$4 are small, of length $\overline{M}_g$5, so one can solve a small linear system for each type and find nonnegative $\overline{M}_g$6.

For $\overline{M}_g$7, the proof is computer-assisted and uses the dual description in terms of pairwise-balanced designs (PBDs). One identifies the cone of effective boundary curves, the PBD cone, inside $\overline{M}_g$8, and then shows by linear-programming and a support-criticality test that every extremal effective curve class is a nonnegative combination of F-curves. Thus the dual cone to the boundary is generated by F-curves, which is equivalent to the Strong F-conjecture.

These two cases are pivotal because they supply the finite input required by the later inductive reduction. They also illustrate the mixed character of the subject: the $\overline{M}_g$9 argument is small-scale and explicit, whereas the $44$0 argument depends on a polyhedral and computational analysis of curve cones.

5. Consequences for $44$1 up to genus $44$2

By the Bridge Theorem of Gibney–Keel–Morrison, the F-conjecture for $44$3 reduces to the symmetric F-conjecture for $44$4, and in particular to the symmetric F-conjecture for $44$5. On the other hand, the inductive strict-partition reduction shows that if the Strong F-conjecture holds for all $44$6 with $44$7, then the symmetric F-conjecture holds for all $44$8 with

$44$9

Since the Strong F-conjecture is now known for $\overline{M}_{g,n}$0, it follows that the symmetric F-conjecture holds for

$\overline{M}_{g,n}$1

Hence by the Bridge Theorem the F-conjecture is true for all $\overline{M}_{g,n}$2 with $\overline{M}_{g,n}$3 in all characteristics $\overline{M}_{g,n}$4 for the GKM reduction (Fedorchuk et al., 16 Jul 2025).

This is one of the most concrete global consequences currently available. Rather than resolving the full conjecture in arbitrary genus, the argument uses genus-$\overline{M}_{g,n}$5 structure to propagate verified cases into a nontrivial genus range. A plausible implication is that further progress on $\overline{M}_{g,n}$6, even for moderately larger $\overline{M}_{g,n}$7, may immediately enlarge the range of genera for which the F-conjecture is known.

6. Failure range and open problems

The Strong F-conjecture does not persist indefinitely. The counterexamples of Pixton show that the Strong F-conjecture fails for $\overline{M}_{g,n}$8, so one cannot push the “complete” boundary-effectivity too far. However, the symmetric F-conjecture itself, namely nef $\overline{M}_{g,n}$9 F-nef for $n$0–invariant divisors, remains open in large $n$1 (Fedorchuk et al., 16 Jul 2025).

This sharply separates the strong and ordinary versions. The strong statement can fail even when the corresponding symmetric nefness problem remains unresolved. A common misconception is therefore to treat the Strong F-conjecture as merely a technical restatement of the F-conjecture; the known failure for $n$2 shows that the two statements have genuinely different logical status.

For $n$3 with $n$4, new ideas will be needed. One may hope for more sophisticated combinatorial reductions, or for discovery of further special F-nef divisors that generate the nef cone. In higher genus the complexity only grows, but the reduction to the symmetric genus-$n$5 case suggests that improved understanding of $n$6, for example via GIT or tropical methods, could eventually settle the full F-conjecture.

7. Terminological ambiguity in other fields

The expression “Strong F-conjecture” is also used in several unrelated areas. This suggests a significant terminological ambiguity across fields.

In graph theory, Dovzhenok–Filuta–Chuhai define the Strong F-Conjecture by the statement: “For every connected graph $n$7 of order $n$8 there exist infinitely many $n$9-irregular graphs.” For graphs $n$00 of diameter $n$01, this conjecture is confirmed by the theorem that for every graph $n$02 with $n$03, there are infinitely many graphs $n$04 with $n$05 such that $n$06 is $n$07-irregular (Dovzhenok, 26 Feb 2026).

In commutative algebra, the related but distinct “Strong Factorial Conjecture” concerns the factorial map

$n$08

on $n$09 and proposes that every polynomial satisfies a weaker vanishing-implies-zero test based on blocks of consecutive powers (Edo et al., 2013).

In additive combinatorics over $n$10, a strong variant of the Polynomial Freiman–Ruzsa conjecture is called the “strong $n$11-conjecture” and asks whether one can choose a linear map $n$12 with

$n$13

for all $n$14; Aaronson exhibited a counterexample to this strong form (Aaronson, 2019).

Within algebraic geometry, however, the Strong F-conjecture ordinarily refers to the boundary-effectivity statement for F-nef divisors on $n$15.