Sagbi bases of Cox–Nagata rings

Abstract

Abstract. 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 Ron, our study emphasizes explicit computations, and offers a new approach to Hilbert functions of fat points.

Presentations of semigroup algebras of weighted trees

“…This multigrade agrees with the one on C[S 1 T ] induced by the level L and the leaf weights under the flat deformation, see [10] for details. Since the multigrading is given by the Picard Group of the blow-up, we refer to it as the Picard grading, and we refer to the associated torus acting on the ring as the Picard torus.…”

“…This gives a common interpretation of both types of semigroups of weighted trees considered here as toric deformations of projective coordinate rings associated to embeddings of blow-ups of projective spaces. It was also noted in [10] that by a theorem of Bauer [2], X (n−3,n) is related to N (0,n) , the moduli space of quasiparabolic semistable rank 2 bundles on P 1 , by a sequence of flops. This implies that these spaces share the same Cox ring, and that the algebras C[S L T (r)] are toric deformations of the projective coordinate rings associated to line bundles on N (0,n) .…”

“…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. When d ≥ 3 the invariant ring R G is isomorphic to the Cox ring of X G , the blow-up of P d−1 at n points determined by G, see [10] for a discussion of this connection. Let K be the pullback of the canonical class on P d−1 , and let the E i be the associated distinguished classes of the blow-up.…”

Finite phylogenetic complexity ofZpand invariants forZ3Michałek1 2017European Journal of Combinatorics2010View full textAdd to dashboardCite

“…Although the original construction of varieties X(T, G) was inspired by phylogenetics, recently they appeared in other sciences [Man09,Man12,Man13,SX10]. We would like also to mention that the varieties X(T, G) share many other very interesting algebraic and combinatorial properties related to their Hilbert polynomial, normality and deformations [BBKM13,BW07,Kub12,MRV14].…”

Zonotopal algebra and forward exchange matroids

“…Federico Ardila and Alex Postnikov studied generalised P-spaces and connections with power ideals [2]. Bernd Sturmfels and Zhiqiang Xu established a connection with Cox rings [51]. Further work on spaces of P-type includes [5,8,39,41,52].…”