Linearizability notions in equivariant birational geometry

Technical Summary: Linearizability Notions in Equivariant Birational Geometry

Problem Statement
This paper investigates the equivariant birational geometry of regular actions of finite groups $G$ on smooth projective varieties over an algebraically closed field $k$ of characteristic zero. The central problem is to understand the relationship between the linearizability of group actions and their behavior under "twisting" via torsors. Specifically, the authors address the extent to which global equivariant properties (such as linearizability, rationality, and unirationality) of a $G$-variety $X$ are determined by the properties of its twists $T_X$ over non-closed fields.

A key tension in the field, as noted in prior work by Duncan and Reichstein ([DR15]), is that while linearizable actions are "versal" (a property related to the existence of rational points in twists), not all versal actions are linearizable. The paper seeks to refine the understanding of these properties by establishing that many global equivariant conditions can be reduced to checking a single, specific twist arising from a generically free versal action, rather than checking all possible torsors.

Methodology
The authors employ a combination of equivariant birational geometry, the theory of torsors, and equivariant intersection theory (Chow groups).

1. Twisting Construction: The core technical tool is the twisting of a $G$-variety $X$ by a $G$-torsor $T = \text{Spec}(L)$ over an extension field $K$. This yields a variety $T_X$ over $K$. The paper leverages the correspondence between $G$-varieties over $k$ and varieties over non-closed fields via this construction.
2. Versality and Generic Freeness: The authors utilize the concept of "versal" $G$-varieties (where every invariant dense open subvariety admits a $G$-equivariant map from any torsor) and "generically free" actions. They focus on constructing a single "test" torsor $T_Z$ arising from a generically free versal $G$-variety $Z$.
3. Equivariant Chow Groups and Diagonal Decomposition: In Section 4, the authors introduce the notion of $G$-equivariant integral decomposition of the diagonal. This involves analyzing the diagonal class $[\Delta_X]$ in the equivariant Chow group $CH^G_n(X \times X)$ and determining if it can be decomposed into a product of the fundamental class and a zero-cycle, plus a term supported on a proper subvariety.
4. Cohomological Invariants: The final section analyzes the behavior of Amitsur invariants and negligible cohomology under twisting, utilizing the Leray spectral sequence and properties of the Brauer group.

Key Contributions and Results

• Reduction to a Single Versal Twist (Theorems 2.5 & 2.6):
  The paper establishes that for a quasi-projective $G$-variety $X$, a wide range of properties holding for all twists $T_X$ (over all torsors $T$) is equivalent to the same property holding for a single twist $T_Z X$, where $Z$ is any quasi-projective $G$-variety with a generically free versal action.

  • Theorem 2.5 covers properties such as the existence of rational points, unirationality, stable rationality, and rationality.
  • Theorem 2.6 extends this equivalence to the birationality of twists ($T_X \sim T_Y$).
  • Significance: This strengthens previous results which required checking properties against twists arising from faithful linear representations. The authors show that any generically free versal action suffices as a "universal test case."

• Coarse G-Birationality and Linearizability (Section 3):
  Motivated by the reduction theorems, the authors define coarse G-birationality ($X \sim^c_G Y$) and coarse linearizability (Condition (CL)).

  • $X$ is coarsely linearizable if it is coarsely G-birational to a projective space $P^d$.
  • This defines a hierarchy: Linearizability (L) $\implies$ Coarse Linearizability (CL) $\implies$ Stable Linearizability (SL).
  • The authors provide examples (e.g., actions of $S_3 \times S_3$ on $P^1 \times P^1$) showing that (CL) is strictly weaker than (L) but strictly stronger than (SL) in certain contexts, offering a finer classification of actions that are not linearizable but behave like linearizable ones under twisting.

• Equivariant Decomposition of the Diagonal (Section 4):
  The authors define $G$-equivariant integral decomposition of the diagonal (Definition 4.2).

  • Theorem 4.4 proves that a $G$-variety $X$ has this property if and only if its twist $T_Z X$ (for a versal $Z$) has the standard (non-equivariant) integral decomposition of the diagonal.
  • Theorem 4.6 connects this to linearizability: If a smooth projective $G$-variety with a generically free action is stably linearizable, it admits a $G$-equivariant integral decomposition of the diagonal.
  • Theorem 4.7 establishes the specialization property: if the general fiber of a smooth proper family of $G$-varieties admits such a decomposition, so does the special fiber.

• Cohomological Obstructions (Section 5):
  The paper investigates the behavior of Amitsur invariants ($Am_d(X, G)$) and the universal torsor obstruction under twisting.

  • Proposition 5.1 states that for weakly versal actions, higher Amitsur invariants vanish.
  • Proposition 5.3 demonstrates that the twisting construction by a versal torsor induces an isomorphism on the Amitsur group $Am_2(X, G)$.
  • Proposition 5.2 characterizes "negligible" cohomology classes as those that vanish when pulled back via the map associated with a single versal torsor, reinforcing the theme that a single versal object controls the cohomological behavior of the entire class of twists.

Significance and Claims
The authors claim that their results represent a strengthening of earlier developments (specifically those in [DR15] and [GM22]). While previous work showed that properties of all twists are implied by properties of twists arising from faithful linear representations, this paper demonstrates that the condition can be relaxed to any generically free versal action.

The paper does not claim to solve the general linearizability problem but rather provides a robust framework for testing equivariant birational properties. By introducing "coarse linearizability" and linking it to the decomposition of the diagonal, the authors offer new tools to distinguish between actions that are genuinely linearizable and those that are only "stably" or "coarsely" so. The work bridges the gap between equivariant geometry over algebraically closed fields and the geometry of varieties over non-closed fields, showing that the latter can often be reduced to the study of a single, well-chosen equivariant model.