isExtremalSymmetricFDivisor -- tests whether an S_n symmetric divisor spans an extremal ray of the cone of symmetric F-divisors

We say a symmetric divisor on $\bar{M}_{0,n}$ is a symmetric F-divisor if $D . F \geq 0$ for every F curve.

Let $SF_{0,n}$ denote the cone of all $S_n$ symmetric divisors on $\bar{M}_{0,n}$ that intersect all the F-curves nonnegatively. This cone contains the cone of $S_n$ symmetric nef divisors. (Fulton's F Conjecture predicts that the two cones are equal). See [AGSS] Section 2 for more details.

This function first checks to see if $D$ is an F-divisor. If not, the function returns false. If so, the function goes on to check whether $D$ is an extremal ray of the cone $SF_{0,n}$. It does so by finding all the F-curves which $D$ intersects in degree zero (i.e., finding how many facets of the cone $D$ lies on) and then checking to see whether this set contains sufficiently many independent hyperplanes to determine an extremal ray.

In the example below, we check that the divisor $3B_2+2B_3+4B_4$ is extremal in the cone $SF_{0,8}$ for $n=8$. We also check that the divisor kappa (see kappaDivisorM0nbar), which is known to be very ample, is not an extremal ray of $SF_{0,8}$.