Rings of polynomial invariants of the alternating group have no finite SAGBI bases with respect to any admissible order

Göbel, Manfred

doi:10.1016/s0020-0190(00)00031-4

Cited by 4 publications

(7 citation statements)

References 5 publications

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“…The proof of Theorem 6.1 below establishes the claimed quadratic presentation for the subspace G which is defined by the specific 3 × 7-matrix A in (15). However, we can multiply each entry of A by a generic complex number, and the initial monomials of the 56 generators will not change.…”

Section: Del Pezzo Surfaces Of Degree One and Twomentioning

confidence: 89%

“…It stands for "subalgebra analogue to Gröbner bases for ideals". Our definition of sagbi bases is more general than the definition usually given in the computer algebra literature [15,19,22]. There one uses a monomial order to define initial monomials and the initial algebra, but this situation can be modeled by introducing an extra variable t as in [13, §15.8] to get to the situation described above.…”

Section: Sagbi Basesmentioning

confidence: 99%

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Sagbi bases of Cox–Nagata rings

Sturmfels

2010

J. Eur. Math. Soc.

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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.

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Section: Del Pezzo Surfaces Of Degree One and Twomentioning

confidence: 89%

Section: Sagbi Basesmentioning

confidence: 99%

Sagbi bases of Cox–Nagata rings

Sturmfels

2010

J. Eur. Math. Soc.

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“…⊂ in > R, and therefore it cannot have a finite SAGBI basis [Robbiano and Sweedler 1990, Example 4.11]. For these and other examples, we refer the reader to [Göbel 2000;Göbel 1999b;Robbiano and Sweedler 1990].…”

Section: When Are Sagbi Bases Finite?mentioning

confidence: 99%

“…For other applications and examples, see [Göbel 2002;2001;2000;1999c;1999b;1999a;Göbel and Maier 2000;Miller 1998;Nordbeck 2002]. …”

Section: Introductionmentioning

confidence: 99%

Some Facts About Canonical Subalgebra Bases

Bravo¹

2004

Trends in Commutative Algebra

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“…To state our last main result, consider the natural (permutation) action of a finite group H ⊂ S n on the polynomial ring [7, 5.6] showed that the invariant ring R = k [x] H has a finite SAGBI basis, with respect to the usual lexicographic term order in k[x], if and only if H = S n1 × · · · × S nr for some partition n 1 + · · · + n r = n. Göbel further conjectured [8, p. 65] that the same should be true for an arbitrary term order in the sense of Definition 1.1 and proved this conjecture in the case where H = A n is the alternating group [9]. In Section 6 we will prove Göbel's conjecture, as an application of our Theorem 1.4:…”

Section: Introductionmentioning

confidence: 99%