F-conjecture in Moduli Spaces

• F-conjecture is a conjectural characterization of the nef cone in the Deligne–Mumford moduli space, determined by nonnegative intersections with all F-curves.
• On M₀,ₙ the conjecture is equivalent to the Mori cone being generated by F-curves, providing a practical numerical test for divisorial positivity.
• Recent results validate the Strong F-conjecture for low n and extend its implications to higher-genus cases, unifying symmetric and non-symmetric formulations.

The F-conjecture is a conjectural description of the nef, and hence dually the Mori, geometry of the Deligne–Mumford moduli space $\overline M_{g,n}$ of stable curves. In its standard form, it predicts that a divisor on $\overline M_{g,n}$ is nef exactly when it has nonnegative intersection with every F-curve, i.e. every one-dimensional boundary stratum. Equivalently, on $\overline M_{0,n}$ one may state that the Mori cone is generated by F-curves and the nef cone coincides with the F-nef cone. Recent work proves that the $S_n$-symmetric and non-symmetric genus-zero formulations are equivalent, establishes the Strong F-conjecture for $\overline M_{0,8}$, and derives the F-conjecture for $\overline M_g$ for all $g\le 44$ (Fedorchuk et al., 16 Jul 2025).

1. Moduli-theoretic setting and F-curves

Let $\overline M_{g,n}$ denote the Deligne–Mumford compactification of the moduli of genus-$g$ curves with $n$ marked points. For a partition

$\overline M_{g,n}$0

the notation $\overline M_{g,n}$1 denotes the corresponding irreducible boundary divisor, namely the closure of the locus of nodal curves whose marked points split as $\overline M_{g,n}$2 and $\overline M_{g,n}$3. In genus zero, these divisors exhaust the boundary. In higher genus, one also has boundary classes $\overline M_{g,n}$4 arising from separating and non-separating nodes (Fedorchuk et al., 16 Jul 2025).

An F-curve is a one-dimensional boundary stratum of $\overline M_{g,n}$5. In genus zero, an F-curve is the image of the unique $\overline M_{g,n}$6 under a gluing morphism attaching fixed stable trees of rational tails. More generally, F-curves are obtained by attaching genus-tails of total genus $\overline M_{g,n}$7 carrying some of the $\overline M_{g,n}$8 markings to the four legs of $\overline M_{g,n}$9. On $\overline M_{0,n}$0 this admits the equivalent description via a $\overline M_{0,n}$1-partition of the markings into four nonempty disjoint subsets, producing a one-parameter family of comb-type stable curves (Fedorchuk et al., 16 Jul 2025).

These curves provide finitely many numerical tests for positivity. A $\overline M_{0,n}$2-Cartier divisor $\overline M_{0,n}$3 on $\overline M_{0,n}$4 is called F-nef if

$\overline M_{0,n}$5

The F-conjecture asserts that these inequalities already characterize nefness.

2. Formulation of the conjecture

The conjecture, attributed in the summary to Gibney–Keel–Morrison (2002), is

$\overline M_{0,n}$6

In particular, the nef cone of $\overline M_{0,n}$7 is predicted to be cut out by the finitely many linear inequalities $\overline M_{0,n}$8 as $\overline M_{0,n}$9 ranges over all F-curves (Fedorchuk et al., 16 Jul 2025).

On $S_n$0 this is equivalent, by duality, to the statement that the Mori cone $S_n$1 is generated by F-curves. The conjecture is therefore simultaneously a description of divisorial positivity and of effective one-cycles. The genus-zero case is structurally central because of bridge results relating it to the higher-genus problem (Moon et al., 2016).

For symmetric divisor classes on $S_n$2, one works in the invariant subspace $S_n$3. If a symmetric divisor is written as

$S_n$4

then for the F-curve associated to a partition $S_n$5 one has

$S_n$6

Thus F-nefness is exactly the nonnegativity of all such expressions (Fedorchuk et al., 16 Jul 2025).

3. Symmetric and non-symmetric variants on $S_n$7

Two genus-zero formulations were traditionally distinguished. The non-symmetric F-conjecture asks whether every F-nef divisor on $S_n$8 is nef. The $S_n$9-symmetric F-conjecture asks the same question after restricting to the invariant subspace $\overline M_{0,8}$0. Prior work treated the symmetric problem as a potentially more tractable approximation to the full conjecture.

A substantial invariant-theoretic reduction of the symmetric problem was obtained in 2016. Using the Kapranov–GIT contraction

$\overline M_{0,8}$1

together with the diagonal divisors $\overline M_{0,8}$2 on the quotient and

$\overline M_{0,8}$3

the pull-back of the basic ample class was identified as

$\overline M_{0,8}$4

where $\overline M_{0,8}$5 is the symmetrized boundary divisor. Any nontrivial integral $\overline M_{0,8}$6-invariant F-nef divisor can then be written uniquely as

$\overline M_{0,8}$7

with rational $\overline M_{0,8}$8, bounds $\overline M_{0,8}$9, and convexity constraints

$\overline M_g$0

Classical invariant theory, via the graphical algebra, converts base-point-freeness and semi-ampleness questions into feasibility of finite rational linear systems in variables $\overline M_g$1, hence into a polyhedral geometry problem (Moon et al., 2016).

This approach led to concrete computational results: for $\overline M_g$2, every integral $\overline M_g$3-invariant F-nef divisor on $\overline M_g$4 was shown to be semi-ample, over $\overline M_g$5; for $\overline M_g$6, every integral $\overline M_g$7-invariant F-nef divisor $\overline M_g$8 satisfies that $\overline M_g$9 is base-point-free, and equivalently for any integral $g\le 44$0-invariant ample divisor $g\le 44$1, $g\le 44$2 is very ample over any algebraically closed field (Moon et al., 2016).

4. Equivalence theorem

The principal theorem of Fedorchuk–Mellit states that for each $g\le 44$3 the following are equivalent:

1. every non-symmetric F-nef divisor on $g\le 44$4 is nef;
2. every $g\le 44$5-invariant F-nef divisor on $g\le 44$6 is nef.

Moreover, by the GKM bridge theorem, these are equivalent to the full F-conjecture in all genera and with any number of markings (Fedorchuk et al., 16 Jul 2025).

The nontrivial direction proceeds by starting from an arbitrary F-nef divisor $g\le 44$7 on $g\le 44$8 and constructing an attaching map

$g\le 44$9

together with a symmetric F-nef divisor $\overline M_{g,n}$0 on $\overline M_{g,n}$1 such that

$\overline M_{g,n}$2

If symmetric F-nef divisors on $\overline M_{g,n}$3 are nef, then $\overline M_{g,n}$4 is nef as a pull-back of a nef divisor. The key new ingredient is a family of F-nef functions on large cyclic groups, called “standard / supertotal functions” in the summary, ensuring that the pull-back of the chosen symmetric divisor recovers the original divisor exactly (Fedorchuk et al., 16 Jul 2025).

The conceptual consequence is that the symmetric problem is not merely a restricted variant. Within the framework supplied by the bridge theorem, symmetric nefness criteria on $\overline M_{g,n}$5 control the full conjecture.

5. The Strong F-conjecture and the cases $\overline M_{g,n}$6

A stronger statement, also attributed to GKM in the summary, is the Strong F-conjecture: every F-nef divisor on $\overline M_{g,n}$7 is a $\overline M_{g,n}$8-linear combination of boundary divisors with nonnegative coefficients. The summary notes that this was known previously for $\overline M_{g,n}$9.

Fedorchuk–Mellit prove that the Strong F-conjecture holds on $g$0, and therefore the usual F-conjecture holds on $g$1. The abstract also records an alternative proof for $g$2 (Fedorchuk et al., 16 Jul 2025).

The verification for $g$3 is described by duality on one-cycles. Any extremal effective one-cycle, equivalently any pairwise-balanced design on $g$4, is shown to be a nonnegative combination of the $g$5 F-curves on $g$6. Concretely, these cycles are encoded as integer vectors in $g$7, and linear programming together with cone-chamber checks is used to verify that the “PBD-cone” is contained in the span of F-curves. All extremal rays are enumerated up to the obvious $g$8-symmetry, and each is shown to lie in the F-cone (Fedorchuk et al., 16 Jul 2025).

This result is significant because it provides a complete strong verification at the first nontrivial case beyond the previously known range and furnishes the input for higher-genus consequences through the bridge mechanism.

6. Consequences for $g$9 and current landscape

The GKM bridge theorem reduces the full F-conjecture for $n$0 with no markings to the symmetric F-conjecture for $n$1. Fedorchuk–Mellit further show that if the Strong F-conjecture holds for all $n$2 with $n$3, then the symmetric F-conjecture holds on all $n$4 with

$n$5

Taking $n$6 yields the symmetric F-conjecture for all $n$7, and hence the F-conjecture for $n$8 for all $n$9, with the summary specifying characteristic $\overline M_{g,n}$00 in the bridge step (Fedorchuk et al., 16 Jul 2025).

This places the 2025 theorem in a broader progression. The 2016 invariant-theoretic work supplied computational confirmation of the $\overline M_{g,n}$01-invariant F-conjecture for $\overline M_{g,n}$02 and semi-ampleness results in that range, while also emphasizing that a conceptual proof in general remained open (Moon et al., 2016). The 2025 equivalence theorem changes the logical structure of the subject: proving the symmetric conjecture is, in fact, equivalent to proving the full non-symmetric genus-zero conjecture, and through the bridge theorem to resolving the conjecture for all $\overline M_{g,n}$03.

Several points that might otherwise be treated as distinct are therefore unified. The genus-zero symmetric problem, the genus-zero non-symmetric problem, and the higher-genus F-conjecture are not separate conjectural layers but equivalent formulations within a single birational-geometric statement. The remaining difficulty lies not in passing from symmetry to generality, but in establishing nefness from F-nefness itself.