Symmetric and non-symmetric F-conjectures are equivalent

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 $g\le 44$.