Geodesic
Sagbi bases of Cox–Nagata rings
Journal of the European Mathematical Society, Tome 12 (2010) no. 2, pp. 429-459.

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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–Iarrobino and Buczyńska–Wiśniewski. Inspired by the zonotopal algebras of Holtz and n s Ron, our study emphasizes explicit computations, and offers a new approach to Hilbert functions of fat points.
DOI : 10.4171/jems/204
Classification : 13-XX, 14-XX, 41-XX, 00-XX
Keywords: Cox ring, del Pezzo surface, phylogenetic variety, fat points, Sagbi basis, Nagata action 
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     title = {Sagbi bases of {Cox{\textendash}Nagata} rings},
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Bernd Sturmfels; Zhiqiang Xu. Sagbi bases of Cox–Nagata rings. Journal of the European Mathematical Society, Tome 12 (2010) no. 2, pp. 429-459. doi : 10.4171/jems/204. http://geodesic.mathdoc.fr/articles/10.4171/jems/204/

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