Symmetric and non-symmetric F-conjectures are equivalent

TL;DR

This paper proves the equivalence of symmetric and non-symmetric F-conjectures for the ample cone of moduli spaces of curves, confirms the Strong F-conjecture for specific cases, and extends the F-conjecture to higher genus.

Contribution

It establishes the equivalence between symmetric and non-symmetric F-conjectures and verifies the Strong F-conjecture for certain moduli spaces, advancing understanding of the ample cone.

Findings

Proved the equivalence of symmetric and non-symmetric F-conjectures.

Confirmed the Strong F-conjecture for 0,8.

Extended the F-conjecture validity up to genus 44.

Abstract

The F-conjecture gives a conjectural description of the ample cone of the Deligne-Mumford moduli space $\overline{M}_{g,n}$. We prove that the $S_n$-symmetric and the non-symmetric F-conjectures are equivalent. We also prove the Strong F-conjecture for $\overline{M}_{0,8}$ (and give an alternative proof for $\overline{M}_{0,7})$. Finally, we derive, as a consequence, the F-conjecture for the moduli space of stable curves $\overline{M}_{g}$ up to genus $g\leq 44$.