Cylinders and the zero locus of the plinth ideal

1. Problem Statement

The paper addresses the geometric characterization of $\mathbb{G}_a$-actions (additive group actions) on affine varieties. Specifically, it seeks to describe the locus of points in an affine variety $X$ where the action admits a "local principal trivialization."

In the context of $\mathbb{G}_a$-actions, which correspond bijectively to Locally Nilpotent Derivations (LNDs) on the coordinate ring $K[X]$, the authors investigate the union of all principal invariant cylinders. A principal cylinder is an open subset $U \subseteq X$ isomorphic to $\mathbb{A}^1 \times Z$ such that the $\mathbb{G}_a$-action acts by translation on the $\mathbb{A}^1$ factor.

The central question is: What is the geometric complement of the union of all such principal cylinders? The paper posits that this complement is exactly the zero locus of the plinth ideal associated with the corresponding LND.

2. Methodology and Definitions

Algebraic Framework

• Setting: $K$ is an algebraically closed field of characteristic zero. $X$ is an affine variety.
• Correspondence: A $\mathbb{G}_a$-action $\alpha$ on $X$ corresponds to an LND $\partial$ on $K[X]$.
  • $\text{Ker}(\partial)$: The ring of invariants.
  • $\text{Im}(\partial)$: The image of the derivation.
• Plinth Ideal: Defined as $\text{pl}(\partial) = \text{Ker}(\partial) \cap \text{Im}(\partial)$. This is an ideal within the ring of invariants.
• Cylinders:
  • A cylinder in $X$ is an open set $U \cong \mathbb{A}^1 \times Z$.
  • An $\alpha$-cylinder is an $\alpha$-invariant cylinder where the action is cylindrical (translation along the $\mathbb{A}^1$ factor).
  • A principal $\alpha$-cylinder is a cylinder of the form $D_X(h) = \{x \in X \mid h(x) \neq 0\}$ for some $h \in K[X]$.

Key Technical Tools

1. Slice Theorem: Used to characterize when an action is cylindrical (existence of a "slice" $s$ such that $\partial(s)=1$).
2. Gluing Derivations: The author uses Lemma 2.1 to show that locally defined LNDs on invariant open sets agree on overlaps, allowing the construction of global functions (like level functions) from local data.
3. Divisor Class Groups: In the counterexamples, the author utilizes the properties of the divisor class group $\text{Cl}(X)$ to determine whether open subsets are principal or affine.

3. Key Contributions and Main Results

Theorem 2.6: Characterization of Principal Cylinders

The paper establishes a precise algebraic condition for a principal open set $D_X(h)$ to be an $\alpha$-cylinder.

• Result: $D_X(h)$ is an $\alpha$-cylinder if and only if $h^n \in \text{pl}(\partial)$ for some integer $n \geq 1$.
• Implication: This links the geometric property of being a cylinder directly to the algebraic structure of the plinth ideal.

Theorem 2.6 (Corollary 2.7): The Main Geometric Description

This is the central theorem of the paper.

• Statement: Let $\{U_i\}_{i \in I}$ be the set of all principal $\alpha$-cylinders in $X$. Then:
  $$X \setminus \bigcup_{i \in I} U_i = V_X(\text{pl}(\partial))$$
• Meaning: The union of all principal invariant cylinders is exactly the complement of the zero locus of the plinth ideal. Points outside this union are precisely those where the action cannot be locally trivialized as a principal cylinder.

Corollary 2.8: Existence of Maximal Cylinders

If the plinth ideal $\text{pl}(\partial)$ is a principal ideal in $\text{Ker}(\partial)$, then there exists a maximal principal $\alpha$-cylinder in $X$. This generalizes previous results known for factorial varieties.

4. Counterexamples and Limitations (Section 3)

The author provides four examples to demonstrate that the description of the union of cylinders cannot be simplified further (e.g., one cannot simply say the union is $X \setminus \text{Fix}(\alpha)$ or that the union is always a single cylinder).

1. Example 3.1 (Non-cylindrical complement):

   • Context: $\mathbb{G}_a$-action on $\mathbb{A}^3$.
   • Finding: The complement of the fixed point set, $\mathbb{A}^3 \setminus \text{Fix}(\alpha)$, is a cylinder topologically but not an $\alpha$-cylinder. The level function required to trivialize the action is not regular on the whole space.

2. Example 3.3 (Non-covering by cylinders):

   • Context: A free action on an affine surface $X \subset \mathbb{P}^2$.
   • Finding: The union of all $\alpha$-cylinders is a proper subset of $X$ (specifically where $z \neq 0$). The surface $X$ cannot be covered by $\alpha$-cylinders, even though the action is free. This relies on the fact that $\text{Cl}(X) \cong \mathbb{Z}/2\mathbb{Z}$, preventing certain open sets from being principal.

3. Example 3.4 (Non-principal covering):

   • Context: An affine $\alpha$-cylinder $U$ inside an affine variety $X$ (Danielewski surface).
   • Finding: There exists an affine $\alpha$-cylinder $U$ that cannot be covered by principal $\alpha$-cylinders. This shows that the union of principal cylinders is strictly smaller than the union of all affine cylinders.

4. Example 3.5 (Non-cylindrical plinth locus):

   • Context: $\mathbb{G}_a$-action on $\mathbb{A}^4$.
   • Finding: The open set $D_X(\text{pl}(\partial))$ (the complement of the zero locus of the plinth ideal) is not an $\alpha$-cylinder. This confirms that while the union of principal cylinders is $D_X(\text{pl}(\partial))$, this set itself does not necessarily possess the structure of a single cylinder.

5. Significance and Impact

• Geometric Interpretation of Algebraic Invariants: The paper provides a concrete geometric meaning to the plinth ideal, a concept previously studied mostly in algebraic terms (e.g., in the context of factoriality). It identifies the plinth ideal's zero locus as the "obstruction set" for principal cylinder trivializations.
• Refinement of $\mathbb{G}_a$-Action Theory: It clarifies the relationship between the fixed point set, the plinth ideal, and the existence of global or local trivializations. It distinguishes between the union of all cylinders and the union of principal cylinders.
• Counterexamples to Intuition: The examples serve as crucial warnings against assuming that:
  • The complement of fixed points is always a cylinder.
  • Free actions imply the variety is covered by cylinders.
  • The set defined by the plinth ideal is itself a cylinder.
• Applications: These results are relevant to the study of automorphism groups of affine varieties and the Zariski Cancellation Problem, areas where $\mathbb{G}_a$-actions and cylinder structures play a pivotal role.

In summary, Shakhmatov successfully bridges the gap between the algebraic definition of the plinth ideal and the geometric structure of $\mathbb{G}_a$-orbits, proving that the "cylindrical part" of an affine variety is precisely the locus where the plinth ideal does not vanish.