---
title: Strong F-conjecture in Moduli of Curves
url: https://www.emergentmind.com/topics/strong-f-conjecture
type: topic
---

# Strong F-conjecture in Moduli of Curves

Searching arXiv for the specified paper and closely related work on the Strong F-conjecture in moduli of curves.
arXiv search query: 2507.12434
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 $44$ [2507.12434].

## 1. Moduli-theoretic setting

Let $\overline{M}_{g,n}$ be the Deligne–Mumford moduli space of stable $n$–pointed curves of genus $g$. Its Picard group is generated over $\mathbb{Q}$ by the boundary divisors $\Delta_{I,J}$ parameterizing nodal curves with markings split into two disjoint subsets $I\sqcup J=[n]$, together with the cotangent-line classes $\psi_i$.

A one-dimensional boundary stratum of $\overline{M}_{g,n}$ is called an F-curve. A divisor $D\in\operatorname{Pic}(\overline{M}_{g,n})\otimes\mathbb{Q}$ is called F-nef if for every F-curve $C\subset\overline{M}_{g,n}$ one has $D\!\cdot C\ge0$.

The F-conjecture for $\overline{M}_{g,n}$ states that a divisor on $\overline{M}_{g,n}$ 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 $0$ and replaces the numerical nefness criterion by a boundary-effectivity statement: every F-nef divisor on $\overline{M}_{0,n}$ is $\mathbb{Q}$–linearly equivalent to an effective combination of boundary divisors [2507.12434].

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 $D$ on $\overline{M}_{0,n}$ as
$$
D \;=\; -\sum_{I\sqcup J=[n]}\,b_{I,J}\,\Delta_{I,J}.
$$
The strong form demands $b_{I,J}\ge0$ after adding relations in $\operatorname{Pic}(\overline{M}_{0,n})$, namely the Keel relations [2507.12434].

An F-curve in $\overline{M}_{0,n}$ corresponds to a partition
$$
[n]=X\sqcup Y\sqcup Z\sqcup W
$$
into four nonempty parts. For such a curve,
$$
D\cdot C_{X,Y,Z}
\;=\;
b_{X,Y\cup Z\cup W}+b_{Y,X\cup Z\cup W}+b_{Z,X\cup Y\cup W}+b_{X\cup Y\cup Z,W}
\;-\;b_{X\cup Y,Z\cup W}-b_{X\cup Z,Y\cup W}-b_{Y\cup Z,X\cup W}\,.
$$
Requiring all these pairings to be nonnegative is the F-nef condition.

A second description writes divisors on $\overline{M}_{0,n}$ in terms of weights $w(i,j)$:
$$
D \;=\; \sum_{1\le i<j\le n}\!\!\!w(i,j)\,\psi_n
\;-\;
\sum_{\substack{I\sqcup J=[n]\\|I|,|J|\ge2}}
\Bigl(\sum_{i\in I,\,j\in J}w(i,j)\Bigr)\,\Delta_{I,J}.
$$
Here $w(i,j)\ge0$ is equivalent to $D$ being an effective boundary. For an F-curve $C_{X,Y,Z,W}$ one has
$$
D\!\cdot C_{X,Y,Z,W}
=\sum_{i\in X,\,j\in Y\cup Z\cup W}w(i,j)\;+\;
\sum_{i\in Y,\,j\in X\cup Z\cup W}w(i,j)\;+\cdots
\;-\;\sum_{i\in X\cup Y,\,j\in Z\cup W}w(i,j)\;-\;\cdots\,.
$$
Checking F-nefness is therefore a finite list of linear inequalities in the $w(i,j)$.

This linear-algebraic reformulation is central to all presently known small-$n$ 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 $S_n$ act by permuting the $n$ markings on $\overline{M}_{0,n}$. One may formulate a weaker symmetric F-conjecture which only tests $S_n$-invariant divisors against all F-curves. Theorem (Fedorchuk–Mellit): the full non-symmetric F-conjecture for all $\overline{M}_{0,n}$ is equivalent to the symmetric F-conjecture. In particular, to prove that every F-nef divisor on $\overline{M}_{0,n}$ is nef it suffices to treat only $S_n$-invariant divisors [2507.12434].

The proof proceeds by transforming an arbitrary F-nef divisor into a pullback of an $S_N$–symmetric F-nef divisor on a larger moduli space. Given any not necessarily symmetric F-nef divisor $D$ on $\overline{M}_{0,n}$, 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 $S_N$–symmetric F-nef divisor on $\overline{M}_{0,N}$. Concretely, one embeds the original labelling set $\{1,\dots,n\}$ into residue classes mod a large integer
$$
N=\prod p_i
$$
so that the pullback of a symmetric function recovers $D$.

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 $0$.

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

A key combinatorial reduction shows that for $S_n$-invariant divisors it suffices to check effectivity on pulling back to boundary strata of type attaching-maps for strict integer partitions
$$
n=a_1+\cdots+a_k
$$
with $a_1>\cdots>a_k$. 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 [2507.12434].

For $\overline{M}_{0,7}$, an alternative proof originally due to Larsen checks by hand that the finitely many $S_7$–invariant F-nef divisors are stratally effective. All possible strict partitions of $7$ are small, of length $\le3$, so one can solve a small linear system for each type and find nonnegative $w(i,j)$.

For $\overline{M}_{0,8}$, 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 $N_1(\overline{M}_{0,8})$, 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 $\overline{M}_{0,7}$ argument is small-scale and explicit, whereas the $\overline{M}_{0,8}$ argument depends on a polyhedral and computational analysis of curve cones.

## 5. Consequences for $\overline{M}_g$ up to genus $44$

By the Bridge Theorem of Gibney–Keel–Morrison, the F-conjecture for $\overline{M}_{g,n}$ reduces to the symmetric F-conjecture for $\overline{M}_{0,g+n}/S_g$, and in particular to the symmetric F-conjecture for $\overline{M}_{0,g}$. On the other hand, the inductive strict-partition reduction shows that if the Strong F-conjecture holds for all $\overline{M}_{0,m}$ with $m\le k$, then the symmetric F-conjecture holds for all $\overline{M}_{0,n}$ with
$$
n \;\le\; \frac{(k+1)(k+2)}2 -1\,.
$$
Since the Strong F-conjecture is now known for $n\le8$, it follows that the symmetric F-conjecture holds for
$$
n\le\frac{9\cdot10}2-1=44.
$$
Hence by the Bridge Theorem the F-conjecture is true for all $\overline{M}_g$ with $g\le44$ in all characteristics $\neq2$ for the GKM reduction [2507.12434].

This is one of the most concrete global consequences currently available. Rather than resolving the full conjecture in arbitrary genus, the argument uses genus-$0$ structure to propagate verified cases into a nontrivial genus range. A plausible implication is that further progress on $\overline{M}_{0,n}$, even for moderately larger $n$, 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 $n\ge12$, so one cannot push the “complete” boundary-effectivity too far. However, the symmetric F-conjecture itself, namely nef $=$ F-nef for $S_n$–invariant divisors, remains open in large $n$ [2507.12434].

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 $n\ge12$ shows that the two statements have genuinely different logical status.

For $\overline{M}_{0,n}$ with $n>44$, 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-$0$ case suggests that improved understanding of $\overline{M}_{0,n}/S_n$, 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 $F$ of order $|F|\ge3$ there exist infinitely many $F$-irregular graphs.” For graphs $F$ of diameter $2$, this conjecture is confirmed by the theorem that for every graph $F$ with $\mathrm{diam}(F)=2$, there are infinitely many graphs $G$ with $\mathrm{diam}(G)=3$ such that $G$ is $F$-irregular [2602.23227].

In commutative algebra, the related but distinct “Strong Factorial Conjecture” concerns the factorial map
$$
\mathcal{L}(X_1^{\ell_1}\cdots X_m^{\ell_m})=\ell_1!\cdots\ell_m!
$$
on $\mathbb{C}[X_1,\dots,X_m]$ and proposes that every polynomial satisfies a weaker vanishing-implies-zero test based on blocks of consecutive powers [1304.3956].

In additive combinatorics over $\mathbb{F}_p$, a strong variant of the Polynomial Freiman–Ruzsa conjecture is called the “strong $F$-conjecture” and asks whether one can choose a linear map $\widetilde\phi$ with
$$
\phi(x)-\widetilde\phi(x)\in tS
$$
for all $x$; Aaronson exhibited a counterexample to this strong form [1902.00353].

Within algebraic geometry, however, the Strong F-conjecture ordinarily refers to the boundary-effectivity statement for F-nef divisors on $\overline{M}_{0,n}$.

Source: https://www.emergentmind.com/topics/strong-f-conjecture