Cohomological obstructions to equivariant unirationality

Technical Summary: Cohomological Obstructions to Equivariant Unirationality

Problem Statement
The paper investigates the problem of determining whether a smooth projective rational variety $X$ with a generically free regular action of a finite group $G$ is $G$-unirational or projectively $G$-unirational.

• $X$ is $G$-unirational if there exists a dominant $G$-equivariant rational map $\mathbb{P}(V) \dashrightarrow X$ for some $G$-representation $V$.
• $X$ is projectively $G$-unirational if such a map exists for a projective representation of $G$.

While the existence of $k$-rational points is a standard obstruction to rationality over non-closed fields, and the existence of fixed points for abelian subgroups (Condition A) is a necessary condition for equivariant unirationality, these obstructions are often insufficient. The authors address the gap where these "obvious" obstructions vanish, yet the variety may still fail to be equivariantly unirational.

Methodology
The authors develop and apply cohomological obstructions derived from the action of $G$ on the Picard group $\text{Pic}(X)$. The core methodology relies on the Leray spectral sequence for the quotient stack $[X/G]$, which yields a sequence of homomorphisms in group cohomology:
$$\text{Pic}(X)^G \xrightarrow{\delta_2} H^2(G, k^\times) \quad \text{and} \quad H^1(G, \text{Pic}(X)) \xrightarrow{\delta_3} H^3(G, k^\times)$$
The images of these maps define the Amitsur groups:

• $Am_2(X, G) := \text{Im}(\delta_2)$
• $Am_3(X, G) := \text{Im}(\delta_3)$

These groups are stable $G$-birational invariants. The paper establishes that:

1. If $X$ is $G$-unirational, then $Am_j(X, H) = 0$ for all subgroups $H \subseteq G$ and $j=2, 3$.
2. If $X$ is projectively $G$-unirational, then $Am_2(X, H)$ must be cyclic and $Am_3(X, H) = 0$.

The authors relate these invariants to the Bogomolov multiplier $B_n(G)$, defined as the kernel of the restriction map from $H^n(G, k^\times)$ to the direct sum of cohomology groups of all abelian subgroups. Specifically, if Condition (A) holds, $Am_j(X, G) \subseteq B_j(G)$. The computation of these invariants is reduced to $p$-Sylow subgroups and utilizes the stabilizer stratification of the $G$-action on $X$.

Key Contributions and Results
The paper provides a classification of generically free regular actions on del Pezzo surfaces and Kummer quartic double solids where these cohomological obstructions are non-trivial, even when Condition (A) is satisfied.

1. Del Pezzo Surfaces (Theorem 3):
   The authors classify pairs $(X, G)$ where $X$ is a del Pezzo surface and $Am_3(X, H) \neq 0$ for some $H \subseteq G$.

   • Degree 3 (Cubic Surfaces): The only case with a non-trivial obstruction satisfying Condition (A) involves the Fermat cubic surface with $G$ containing $H = C_3^3$. However, this case fails Condition (A) for the full group, though the obstruction persists.
   • Degree 2: The authors identify specific actions where Condition (A) holds (fixed points exist for all abelian subgroups), yet $Am_3(X, H) \neq 0$.
     • The primary obstruction arises from actions of the quaternion group $Q_8$ (order 8).
     • Specifically, for a degree 2 del Pezzo surface defined by $w^2 = x_1^4 + x_2^4 + x_3^4 + ax_1^2x_2^2$, the action of $Q_8$ generated by specific involutions and permutations yields $Am_3(X, Q_8) \cong \mathbb{Z}/2$.
     • This result demonstrates that the analog of a known theorem for higher-degree del Pezzo surfaces (where Condition (A) implies unirationality) fails for degree 2 surfaces.

2. Kummer Quartic Double Solids (Theorem 4):
   The study extends to double covers of $\mathbb{P}^3$ ramified in a Kummer surface (arising from the Jacobian of a genus 2 curve with maximal automorphisms).

   • The authors compute $Am_3(\tilde{X}, G)$ for the blow-up $\tilde{X}$ of the 16 nodes.
   • They identify that $Am_3(\tilde{X}, H) \neq 0$ if and only if $G$ contains a subgroup conjugate to a specific $Q_8$ action.
   • Unlike the surface case, the computation reveals that a non-trivial Brauer group of the quotient stack can exist even in the absence of divisors with non-trivial generic stabilizers, requiring the analysis of higher-dimensional strata.

Significance and Claims
The paper claims to provide new cohomological obstructions that are distinct from the existence of fixed points.

• Subtlety of Equivariant Geometry: The results highlight a "stark difference" between birational geometry over a field $k$ (where rational points are the primary obstruction) and equivariant geometry over the stack $BG$. In the equivariant setting, invariants like $Am_3$ are "quite subtle and informative."
• Failure of Condition (A): The paper explicitly shows that for del Pezzo surfaces of degree 2, the existence of fixed points for all abelian subgroups (Condition A) is not sufficient to guarantee $G$-unirationality or projective $G$-unirationality.
• Role of $Q_8$: In all identified counterexamples, the quaternion group $Q_8$ is the source of the non-trivial obstruction.
• Conclusion: Varieties exhibiting these non-trivial Amitsur invariants are neither $G$-unirational nor projectively $G$-unirational, providing a concrete method to exclude equivariant unirationality in cases where geometric fixed-point criteria are met.