Sagbi Bases of Cox-Nagata Rings

Details Sagbi Bases of Cox-Nagata Rings Sagbi Bases of Cox-Nagata Rings

2008-03-06

arxiv.org/abs/0803.0892v2

Mathematics

Algebraic Geometry

35 pages. Results in Section 7 improved

Scientific paper

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.

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