[Paper Review] Factorial algebraic group actions and categorical quotients

Ivan Arzhantsev, Devrim Celik+1 more|arXiv (Cornell University)|Aug 4, 2009

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper establishes conditions for the existence of categorical quotients in algebraic geometry under factorial group actions, introducing a novel framework using constructible spaces to overcome limitations in the classical variety category. It proves that for factorial actions of affine algebraic groups on normal varieties, a categorical quotient exists in the category of constructible spaces if and only if the invariant ring satisfies a codimension condition, generalizing GIT-type constructions for complete varieties with finitely generated Cox rings.

ABSTRACT

Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result which, for example, settles the case of a finitely generated algebra of invariants. As an application, we provide a combinatorial GIT-type construction of categorial quotients for actions on, e.g. complete varieties with finitely generated Cox ring via lifting to the characteristic space.

Motivation & Objective

• To address the lack of categorical quotients in the category of algebraic varieties when the group is not reductive or the variety is not affine.
• To extend the theory of categorical quotients beyond the classical reductive case by introducing constructible spaces as a more flexible quotient category.
• To provide a combinatorial, GIT-type construction of categorical quotients for actions on complete varieties with finitely generated Cox rings via lifting to the characteristic space.
• To characterize the existence of categorical quotients in the category of varieties using finitely generated separating subalgebras of invariants with open image.

Proposed method

• Introduce the concept of factorial group actions, where every invariant hypersurface is the zero set of an invariant function, generalizing the reductive case.
• Work in the category of constructible spaces, allowing quotients to be constructible subsets of affine varieties, thus relaxing the requirement for the quotient to be a variety.
• Prove that a morphism π:X→Y is a strong categorical quotient in the constructible category if and only if the pullback π∗:Γ(Y,𝒪)→Γ(X,𝒪)G is an isomorphism.
• Establish a criterion for existence of a categorical quotient in the category of varieties: the canonical map X→Spec(A) must have open image for some finitely generated normal subalgebra A⊆Γ(X,𝒪)G with field of fractions 𝕂(X)G.
• Leverage the existence of separating subalgebras of invariants (from Derksen and Kemper) to construct such A and ensure open image under suitable conditions.
• Apply the theory to complete varieties with finitely generated Cox rings by lifting the G-action to the characteristic space, using torus actions and GIT-fans to compute semistable loci and quotients.

Experimental results

Research questions

• RQ1 Under what conditions does a categorical quotient exist in the category of algebraic varieties for a factorial action of an affine algebraic group?
• RQ2 Can categorical quotients be constructed in a broader category than varieties when the invariant ring is not finitely generated?
• RQ3 How can one systematically construct categorical quotients for actions on complete varieties with finitely generated Cox rings?
• RQ4 What is the role of separating subalgebras of invariants in ensuring the existence of categorical quotients?
• RQ5 How does the combinatorics of the GIT-fan for a torus action on the characteristic space relate to the variation of categorical quotients?

Key findings

• A categorical quotient exists in the category of constructible spaces if and only if the pullback π∗:Γ(Y,𝒪)→Γ(X,𝒪)G is an isomorphism, which holds precisely when the image of X→Spec(A) is open and its complement has codimension at least two.
• When the invariant ring Γ(X,𝒪)G is finitely generated, the canonical map X→Y with Y=π′(X) is a strong categorical quotient in the category of constructible spaces.
• For factorial actions, the existence of a categorical quotient in the category of varieties is equivalent to the existence of a finitely generated normal separating subalgebra A⊆Γ(X,𝒪)G such that X→Spec(A) has open image.
• The theory enables a combinatorial GIT-type construction of categorical quotients for complete varieties with finitely generated Cox rings by lifting the G-action to the characteristic space and analyzing the induced torus action.
• In the example of the additive group acting on P¹×P¹, two semistable loci U₁ and U₂ admit strong categorical quotients Uᵢ→P¹, corresponding to the two full-dimensional chambers in the GIT-fan.
• The construction yields a strong categorical quotient X^{ss}(D)→V for any ample divisor D, provided the semistable locus in the characteristic space satisfies the codimension condition.