[Paper Review] Makar-Limanov invariant, Derksen invariant, flexible points

Fabrizio Donzelli|arXiv (Cornell University)|Jul 17, 2011

Advanced Differential Equations and Dynamical Systems4 references3 citations

TL;DR

This paper establishes a deep connection between the Makar-Limanov invariant, the Derksen invariant, and the existence of flexible points on affine varieties. It proves that for affine varieties with finite Picard group, the Makar-Limanov invariant is trivial (isomorphic to ℂ) if and only if a flexible point exists, and further shows that trivial Derksen invariants imply transitive SAut(X) action with at most one fixed point, providing a structural classification tool for algebraic automorphism groups in affine algebraic geometry.

ABSTRACT

We estabilish some connections among the Makar-Limanov invariant, the Derksen invariant, and the existence of flexible points on an affine variety.

Motivation & Objective

• To clarify the relationship between the Makar-Limanov invariant, the Derksen invariant, and the geometric notion of flexibility in affine algebraic varieties.
• To resolve a gap in the literature by showing that the triviality of the regular Makar-Limanov invariant implies the existence of flexible points under finite Picard group conditions.
• To provide a structural characterization of affine surfaces with trivial invariants, particularly in the context of Danilov-Gizatullin surfaces and their automorphism groups.
• To prove that a trivial Derksen invariant implies that the automorphism group has at most one fixed point, offering a new invariant-based classification criterion.

Proposed method

• Uses the special automorphism group SAut(X), generated by all ℂ+-actions, to analyze orbit structures and transitivity on the regular locus of X.
• Applies the concept of locally nilpotent derivations (LNDs) to define the Makar-Limanov invariant as the intersection of kernels of all LNDs on X.
• Employs a geometric quotient construction via rational SAut(X)-invariants to build an open affine subset X′ with a geometric quotient X′//SAut(X).
• Utilizes a lemma stating that if a locally nilpotent derivation is tangent to a hypersurface D, then the defining function g of D is invariant under the derivation.
• Applies a generalization of Rosenlicht’s theorem on rational invariants to show the existence of a finite set of rational invariants that separate orbits in general position.
• Performs direct computation of the Derksen invariant for Danilov-Gizatullin surfaces using explicit locally nilpotent derivations and monomial generators of the coordinate ring.

Experimental results

Research questions

• RQ1 Under what conditions does the triviality of the Makar-Limanov invariant imply the existence of a flexible point on an affine variety?
• RQ2 How are the Derksen invariant and the existence of flexible points related in the context of affine surfaces with finite Picard group?
• RQ3 Can the automorphism group of an affine variety with trivial Derksen invariant have more than one fixed point?
• RQ4 What is the precise relationship between the field Makar-Limanov invariant and the existence of flexible points, especially in singular or non-smooth cases?
• RQ5 Do Danilov-Gizatullin surfaces have trivial Derksen invariants, and can this be verified via direct computation of the invariant using explicit derivations?

Key findings

• The field Makar-Limanov invariant FML(X) is trivial (isomorphic to ℂ) if and only if X has a flexible point, and this is equivalent to SAut(X) acting with an open orbit.
• For affine varieties with finite Picard group, the regular Makar-Limanov invariant ML(X) is trivial if and only if there exists a flexible point on X.
• The Derksen invariant D(X) is equal to ℂ[X] if and only if the automorphism group Aut(X) has at most one fixed point.
• For Danilov-Gizatullin surfaces V_n, both the Makar-Limanov and Derksen invariants are trivial: ML(V_n) ≅ ℂ and D(V_n) = ℂ[V_n].
• Direct computation confirms that the Derksen invariant of V_n is generated by monomials x_k and y, and that the invariants span a vector space of dimension b+1, proving triviality.
• The paper shows that if the complement of the open Aut(X)-orbit on a surface has at least two points and ML(X) ≅ ℂ, then D(X) ≠ ℂ[X], providing a criterion for non-trivial Derksen invariants.