---
title: F-conjecture in Moduli Spaces
url: https://www.emergentmind.com/topics/f-conjecture
type: topic
---

# F-conjecture in Moduli Spaces

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 $g\le 44$ [2507.12434].

## 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
\[
I\sqcup J=\{1,\dots,n\},\qquad 2\le |I|,|J|\le n-2,
\]
the notation $\Delta_I\equiv \Delta_{I,J}\subset \overline M_{g,n}$ denotes the corresponding irreducible boundary divisor, namely the closure of the locus of nodal curves whose marked points split as $I$ and $J$. In genus zero, these divisors exhaust the boundary. In higher genus, one also has boundary classes $\delta_0,\dots,\delta_{\lfloor g/2\rfloor}$ arising from separating and non-separating nodes [2507.12434].

An **F-curve** is a one-dimensional boundary stratum of $\overline M_{g,n}$. In genus zero, an F-curve is the image of the unique $\overline M_{0,4}$ under a gluing morphism attaching fixed stable trees of rational tails. More generally, F-curves are obtained by attaching genus-tails of total genus $g$ carrying some of the $n$ markings to the four legs of $\overline M_{0,4}$. On $\overline M_{0,n}$ this admits the equivalent description via a $4$-partition of the markings into four nonempty disjoint subsets, producing a one-parameter family of comb-type stable curves [2507.12434].

These curves provide finitely many numerical tests for positivity. A $\mathbb Q$-Cartier divisor $D$ on $\overline M_{g,n}$ is called **F-nef** if
\[
D\cdot F\ge 0
\qquad\text{for every F-curve }F\subset \overline M_{g,n}.
\]
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
\[
\boxed{
\text{A divisor }D\in\Pic(\overline M_{g,n})_\mathbb R
\text{ is nef}
\Longleftrightarrow
D\text{ is F-nef.}
}
\]
In particular, the nef cone of $\overline M_{g,n}$ is predicted to be cut out by the finitely many linear inequalities $D\cdot F\ge 0$ as $F$ ranges over all F-curves [2507.12434].

On $\overline M_{0,n}$ this is equivalent, by duality, to the statement that the Mori cone $NE_1(\overline M_{0,n})$ 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 [1606.02232].

For symmetric divisor classes on $\overline M_{0,n}$, one works in the invariant subspace $\Pic(\overline M_{0,n})^{S_n}$. If a symmetric divisor is written as
\[
D=a\,\psi-\sum_{I\subset[n],\,|I|\ge 2} b_{|I|}\,\Delta_I,
\]
then for the F-curve associated to a partition $\{1,\dots,n\}=I\sqcup J\sqcup K\sqcup L$ one has
\[
D\!\cdot\! F_{I,J,K,L}
=
b_{|I\cup J|}+b_{|I\cup K|}+b_{|I\cup L|}
-
\bigl(b_{|I|}+b_{|J|}+b_{|K|}+b_{|L|}\bigr).
\]
Thus F-nefness is exactly the nonnegativity of all such expressions [2507.12434].

## 3. Symmetric and non-symmetric variants on $\overline M_{0,n}$

Two genus-zero formulations were traditionally distinguished. The **non-symmetric F-conjecture** asks whether every F-nef divisor on $\overline M_{0,n}$ is nef. The **$S_n$-symmetric F-conjecture** asks the same question after restricting to the invariant subspace $\Pic(\overline M_{0,n})^{S_n}$. 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
\[
T:\overline M_{0,n}\to (\mathbb P^1)^n // SL_2,
\]
together with the diagonal divisors $D_{ij}$ on the quotient and
\[
D_2=\sum_{1\le i<j\le n} D_{ij},
\]
the pull-back of the basic ample class was identified as
\[
T^*D_2=\sum_{i=2}^{\lfloor n/2\rfloor} \binom{i}{2}\,B_i,
\]
where $B_i=\sum_{|I|=i} B_I$ is the symmetrized boundary divisor. Any nontrivial integral $S_n$-invariant F-nef divisor can then be written uniquely as
\[
D=T^*(cD_2)-\sum_{i=3}^{\lfloor n/2\rfloor} a_i B_i,
\]
with rational $c>0$, bounds $0\le a_i<c\binom{i}{2}$, and convexity constraints
\[
a_j+a_{j+2}-2a_{j+1}\ge 0.
\]
Classical invariant theory, via the graphical algebra, converts base-point-freeness and semi-ampleness questions into feasibility of finite rational linear systems in variables $w_{ij}$, hence into a polyhedral geometry problem [1606.02232].

This approach led to concrete computational results: for $n\le 19$, every integral $S_n$-invariant F-nef divisor on $\overline M_{0,n}$ was shown to be semi-ample, over $\operatorname{Spec}\mathbb Z$; for $n\le 16$, every integral $S_n$-invariant F-nef divisor $D$ satisfies that $2D$ is base-point-free, and equivalently for any integral $S_n$-invariant ample divisor $A$, $2A$ is very ample over any algebraically closed field [1606.02232].

## 4. Equivalence theorem

The principal theorem of Fedorchuk–Mellit states that for each $n\ge 4$ the following are equivalent:

1. every non-symmetric F-nef divisor on $\overline M_{0,n}$ is nef;
2. every $S_n$-invariant F-nef divisor on $\overline M_{0,n}$ 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 [2507.12434].

The nontrivial direction proceeds by starting from an arbitrary F-nef divisor $D=D(f)$ on $\overline M_{0,n}$ and constructing an attaching map
\[
\iota:\overline M_{0,n}\longrightarrow \overline M_{0,N},
\qquad N\gg n,
\]
together with a symmetric F-nef divisor $\widetilde D$ on $\overline M_{0,N}$ such that
\[
D=\iota^*(\widetilde D).
\]
If symmetric F-nef divisors on $\overline M_{0,N}$ are nef, then $D$ 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 [2507.12434].

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 $\overline M_{0,n}$ control the full conjecture.

## 5. The Strong F-conjecture and the cases $n=7,8$

A stronger statement, also attributed to GKM in the summary, is the **Strong F-conjecture**: every F-nef divisor on $\overline M_{0,n}$ is a $\mathbb Q$-linear combination of boundary divisors with nonnegative coefficients. The summary notes that this was known previously for $n\le 7$.

Fedorchuk–Mellit prove that the Strong F-conjecture holds on $\overline M_{0,8}$, and therefore the usual F-conjecture holds on $\overline M_{0,8}$. The abstract also records an alternative proof for $\overline M_{0,7}$ [2507.12434].

The verification for $\overline M_{0,8}$ is described by duality on one-cycles. Any extremal effective one-cycle, equivalently any pairwise-balanced design on $\{1,\dots,7\}$, is shown to be a nonnegative combination of the $280$ F-curves on $\overline M_{0,8}$. Concretely, these cycles are encoded as integer vectors in $\mathbb Z^{2^{7}-1}$, 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 $S_8$-symmetry, and each is shown to lie in the F-cone [2507.12434].

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 $\overline M_g$ and current landscape

The GKM bridge theorem reduces the full F-conjecture for $\overline M_g$ with no markings to the symmetric F-conjecture for $\overline M_{0,g}$. Fedorchuk–Mellit further show that if the Strong F-conjecture holds for all $\overline M_{0,m}$ with $m\le k$, then the symmetric F-conjecture holds on all $\overline M_{0,n}$ with
\[
n\le \frac{(k+1)(k+2)}2-1.
\]
Taking $k=8$ yields the symmetric F-conjecture for all $n\le 44$, and hence the F-conjecture for $\overline M_g$ for all $g\le 44$, with the summary specifying characteristic $\ne 2$ in the bridge step [2507.12434].

This places the 2025 theorem in a broader progression. The 2016 invariant-theoretic work supplied computational confirmation of the $S_n$-invariant F-conjecture for $n\le 19$ and semi-ampleness results in that range, while also emphasizing that a conceptual proof in general remained open [1606.02232]. 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 $\overline M_{g,n}$.

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.

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