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

TL;DR

This paper discovers numerous new extreme divisors on moduli space $ar{M}_{0,7}$, highlights differences in the effective cone over characteristic 2 versus characteristic 0, and applies these findings to specific moduli spaces.

Contribution

It identifies thousands of new extreme divisors on $ar{M}_{0,7}$, demonstrates characteristic-dependent differences in the effective cone, and provides explicit cycles illustrating these phenomena.

Findings

Found 101,052 new extreme divisors on $ar{M}_{0,7}$

Proved $ar{ ext{Eff}}^k (ar{M}_{0,n})$ is larger over characteristic 2

Computed $ ext{Eff}(ar{M}_{0, ext{A}})$ as polyhedral over any field

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 $\bar{\text{Eff}}^k (\bar{M}_{0,n})$ is strictly larger over characteristic 2 than it is over characteristic 0, for all $1\leq k \leq n-6$. 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 $\bar{\text{Eff}}^k(\bar{M}_{0,n})$ 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 $\text{Eff}(\text{Bl}_e \bar{LM}_7)$.