  Equations of hypertree divisors on the Grothendieck-Knudsen moduli space of
stable rational curves, introduced by Castravet and Tevelev, appear as
numerators of scattering amplitude forms for n massless particles in N=4
Yang-Mills theory in the work of Arkani-Hamed, Bourjaily, Cachazo, Postnikov
and Trnka. Rather than being a coincidence, this is just the tip of the iceberg
of an exciting relation between algebraic geometry and high energy physics. We
interpret leading singularities of scattering amplitude forms of massless
particles as probabilistic Brill-Noether theory: the study of statistics of
images of n marked points under a random meromorphic function uniformly
distributed with respect to the translation-invariant volume form of the
Jacobian. We focus on the maximum helicity violating regime, which leads to a
beautiful physics-inspired geometry for various classes of algebraic curves:
smooth, stable, hyperelliptic, real algebraic, etc.
