Sagbi Bases of Cox-Nagata Rings

By Bernd Sturmfels, Zhiqiang Xu. 2008

89 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/be3c6cbf02bcdeedcc1f58f690a9728b0c7c612c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

On Cox rings of K3 surfaces (2009, unknown): “A G ] 3 S ep 2 00 9”
Combinatorics and geometry of power ideals (2008, methodology): “Sturmfels and Xu [22] define the zonotopal Cox ring of A to be Z(A) = ⊕”
On the Weak Lefschetz Property for Powers of Linear Forms (2010, background): “For example, their Hilbert functions have been the focus of the papers [1], [23], [14], among others.”