  We degenerate Cox-Nagata rings to toric algebras by means of sagbi bases
induced by configurations over the rational function field. For del Pezzo
surfaces, this degeneration implies the Batyrev-Popov conjecture that these
rings are presented by ideals of quadrics. For the blow-up of projective
n-space at n+3 points, sagbi bases of Cox-Nagata rings establish a link between
the Verlinde formula and phylogenetic algebraic geometry, and we use this to
answer questions due to D'Cruz-Iarobbino and Buczynska-Wisniewski. Inspired by
the zonotopal algebras of Holtz and Ron, our study emphasizes explicit
computations, and offers a new approach to Hilbert functions of fat points.
