Hilbert's 14th problem and Cox rings

Abstract

Our main result is the description of generators of the total coordinate ring of the blow-up of P n in any number of points that lie on a rational normal curve. As a corollary we show that the algebra of invariants of the action of a two-dimensional vector group introduced by Nagata is finitely generated by certain explicit determinants. We also prove the finite generation of the algebras of invariants of actions of vector groups related to T-shaped Dynkin diagrams introduced by Mukai.

Sagbi bases of Cox–Nagata rings

“…We resolve a problem left open by Buczyńska and Wiśniewski in [6], by showing that the phylogenetic varieties in [6,32] arise as flat limits of the Cox-Nagata rings constructed by Castravet and Tevelev in [8]. We also prove a conjecture of D'Cruz and Iarrobino [10] on Hilbert functions of fat points.…”

“…Some relevant references are [3,8,12,21,24,29,31,33]. These papers are primarily concerned with the case when 1 , .…”

Presentations of semigroup algebras of weighted trees

“…The structure of this ring was studied by Castravet and Tevelev in [4] and by Mukai in [7]. Let R be a polynomial ring over C in 2n variables, and let G be a subspace of C n of codimension d. There is an action of G on R which generalizes the construction used by Nagata in his solution to Hilbert's 14th problem.…”

Cox rings of degree one del Pezzo surfaces

“…Cox rings are finitely generated k-algebras in several other cases, including del Pezzo surfaces [Batyrev and Popov 2004], rational surfaces with big anticanonical divisor [Testa et al 2009], blow-ups of ‫ސ‬ n at points lying on a rational normal curve [Castravet and Tevelev 2006] and wonderful varieties [Brion 2007]. All these varieties are examples of Mori dream spaces [Hu and Keel 2000], and for this class the Cox ring of X captures much of the birational geometry of the variety.…”