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

Sagbi bases of Cox–Nagata rings