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Sagbi bases of Cox–Nagata rings

Bernd SturmfelsZhiqiang Xu — 2010

Journal of the European Mathematical Society

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....

Binomial residues

Eduardo CattaniAlicia DickensteinBernd Sturmfels — 2002

Annales de l’institut Fourier

A binomial residue is a rational function defined by a hypergeometric integral whose kernel is singular along binomial divisors. Binomial residues provide an integral representation for rational solutions of A -hypergeometric systems of Lawrence type. The space of binomial residues of a given degree, modulo those which are polynomial in some variable, has dimension equal to the Euler characteristic of the matroid associated with A .

Brownian motion tree models are toric

Bernd SturmfelsCaroline UhlerPiotr Zwiernik — 2020


Felsenstein's classical model for Gaussian distributions on a phylogenetic tree is shown to be a toric variety in the space of concentration matrices. We present an exact semialgebraic characterization of this model, and we demonstrate how the toric structure leads to exact methods for maximum likelihood estimation. Our results also give new insights into the geometry of ultrametric matrices.

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