Extreme Divisors on ${\bar{M}}_{0,7}$ and Differences over Characteristic 2

Abstract

We find 101,052 new extreme divisors on ${\bar{M}}_{0,7}$ (in 31 $S_7$-orbits) and millions of extreme nef curves over characteristic 0. Over characteristic 2, we identify two more $S_7$-orbits of extreme divisors, and prove ${\overline{\text{Eff}}}^k({\bar{M}}_{0,n})$ is strictly larger over characteristic 2 than it is over characteristic 0, for all $1\le k\le n-6$. For each such $k$ we provide explicit cycles which are extreme in ${\text{Eff}}^k({\bar{M}}_{0,n})$ over characteristic 2 but external to ${\overline{\text{Eff}}}^k({\bar{M}}_{0,n})$ over characteristic 0. We apply our method of finding new extreme divisors to compute $\text{Eff}({\bar{M}}_{0,\mathcal{A}})$ for $\mathcal{A}=(\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1)$, proving it is polyhedral over any field, and conjecture a description of $\text{Eff}({\text{Bl}}_e{\overline{LM}}_7)$.

73 pages, 5 figures