On some properties of elementary derivations in dimension six

Khoury, Joseph

doi:10.1016/s0022-4049(99)00121-8

Cited by 12 publications

(4 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In (6), the authors used an induction hypothesis to assume that the kernel D 0 of δ has the form described in the theorem. Since we are using a specific value for m, we can use a result from (4) to assume that D 0 = k[X 1 , X 2 , X 3 , X 4 , X t+1 i Y j − X t+1 j Y i : 1 ≤ i < j ≤ 3] and the isomorphism…”

Section: On the Proof Of (6)mentioning

confidence: 99%

A Groebner basis approach to solve a Conjecture of Nowicki

Khoury¹

2008

Journal of Symbolic Computation

Self Cite

View full text Add to dashboard Cite

Let k be a field of characteristic zero, n any positive integer and let δn be the derivation n i=1 Xi ∂ ∂Y i of the polynomial ring k[X1,. .. , Xn, Y1,. .. , Yn] in 2n variables over k. A Conjecture of Nowicki (Conjecture 6.9.10 in (8)) states the following ker δn = k[X1,. .. , Xn, XiYj − XjYi; 1 ≤ i < j ≤ n] in which case we say that δn is standard. In this paper, we use the elimination theory of Groebner bases to prove that Nowicki's conjecture holds in the more general case of the derivation D = n i=1 X t i i ∂ ∂Y i , ti ∈ Z ≥0. In (6), H. Kojima and M. Miyanishi argued that D is standard in the case where ti = t (i = 1,. .. n) for some t ≥ 3. Although the result is true, we show in Section 4 of this paper that the proof presented in (6) is not complete.

show abstract

Section: On the Proof Of (6)mentioning

confidence: 99%

A Groebner basis approach to solve a Conjecture of Nowicki

Khoury¹

2008

Journal of Symbolic Computation

Self Cite

View full text Add to dashboard Cite

show abstract

“…In this section, let A = k [x, s, t, u, v, z] = k [6] , and define the triangular derivation d on A by…”

Section: A Linear Counterexample In Dimension Elevenmentioning

confidence: 99%

A linear counterexample to the Fourteenth Problem of Hilbert in dimension eleven

Freudenburg

2006

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. A family of G a -actions on affine space A m is constructed, each having a non-finitely generated ring of invariants (m ≥ 6). Because these actions are of small degree, they induce linear actions of unipotent groups G n a G a on A 2n+3 for n ≥ 4, and these invariant rings are also non-finitely generated. The smallest such action presented here is for the group G 4 a G a acting linearly on A 11 .

show abstract

“…Non-linear actions of G a on affine space, and their corresponding locally nilpotent derivations, have been an active area of interest for many years [vdE00]. Elementary monomial derivations have in particular received special attention; several properties that guarantee finite generation of the kernel, and also that guarantee non-finite generation, have been found [Kho01,Kho04,Kur04,Tan07,Kur09]. Thus Corollary 1.3 provides a novel method of approach and places the Hu-Keel question in the context of an extensive body of literature.…”

mentioning

confidence: 99%

Projective linear configurations via non-reductive actions

Doran,

Giansiracusa

2014

Preprint

View full text Add to dashboard Cite

We study the iterated blow-up X of projective space along an arbitrary collection of linear subspaces. By replacing the universal torsor with an A 1 -homotopy equivalent model, built from A 1 -fiber bundles not just algebraic line bundles, we construct an "algebraic uniformization": X is a quotient of affine space by a solvable group action. This provides a clean dictionary, using a single coordinate system, between the algebra and geometry of hypersurfaces: effective divisors are characterized via toric and invariant-theoretic techniques. In particular, the Cox ring is an invariant subring of a Pic(X)-graded polynomial ring and it is an intersection of two explicit finitely generated rings. When all linear subspaces are points, this recovers a theorem of Mukai while also giving it a geometric proof and topological intuition. Consequently, it is algorithmic to describe Cox(X) up to any degree, and when Cox(X) is finitely generated it is algorithmic to verify finite generation and compute an explicit presentation. We consider in detail the special case of M 0,n. Here the algebraic uniformization is defined over Spec Z. It admits a natural modular interpretation, and it yields a precise sense in which M 0,n is "one non-linearizable Ga away" from being a toric variety. The Hu-Keel question becomes a special case of Hilbert's 14th problem for Ga.

show abstract

On some properties of elementary derivations in dimension six

Cited by 12 publications

References 11 publications

A Groebner basis approach to solve a Conjecture of Nowicki

A Groebner basis approach to solve a Conjecture of Nowicki

A linear counterexample to the Fourteenth Problem of Hilbert in dimension eleven

Projective linear configurations via non-reductive actions

Contact Info

Product

Resources

About