[Paper Review] p-elementary subgroups of the Cremona group

Arnaud Beauville|ArXiv.org|Feb 7, 2005

Homotopy and Cohomology in Algebraic Topology4 references4 citations

TL;DR

This paper classifies $p$-elementary subgroups of the Cremona group over an algebraically closed field $k$, specifically those isomorphic to $(\mathbb{Z}/p)^r$ with $p \neq \mathrm{char}(k)$. Using geometric realization on rational surfaces and classification of group actions on del Pezzo surfaces or $\mathbb{P}^1$-fibrations, it shows that for $p \geq 5$, $r \leq 2$, and such subgroups are conjugate to $p$-torsion of diagonal tori in $PGL_3(k)$; for $p=3$, $r \leq 3$, conjugate to $3$-torsion in $PGL_4(k)$ acting on the Fermat cubic; for $p=2$, $r \leq 4$, with explicit conjugacy classes for $r=4$. The key contribution is a complete classification of these subgroups up to conjugacy.

ABSTRACT

We classify, up to conjugacy, the subgroups of the Cremona group isomorphic to (Z/p)^r, where p is prime and r is maximal.

Motivation & Objective

• To classify $p$-elementary subgroups of the Cremona group $\mathrm{Cr}_k$, i.e., subgroups isomorphic to $(\mathbb{Z}/p)^r$ for prime $p \neq \mathrm{char}(k)$, up to conjugacy.
• To extend prior work on finite subgroups of the Cremona group by focusing on $p$-elementary structures, particularly for $p=2,3,5$ and above.
• To determine the maximal possible rank $r$ of such subgroups and describe their conjugacy classes via geometric realizations on rational surfaces.
• To resolve the non-uniqueness of maximal tori containing $p$-elementary subgroups in the Cremona group, contrasting with semisimple groups.

Proposed method

• Realize $p$-elementary subgroups as automorphism groups of minimal rational surfaces $S$ via $G \subset \mathrm{Aut}(S)$, using birational equivalence to $\mathbb{P}^2_k$.
• Reduce the classification to two cases: $G$ preserving a $\mathbb{P}^1$-fibration or $\mathrm{rk}\,\mathrm{Pic}(S)^G = 1$ (i.e., $S$ a del Pezzo surface).
• Classify $p$-elementary subgroups of $\mathrm{Aut}(\mathbb{P}^1_K)$ for function fields $K$ using known results on $PGL_2(K)$, particularly for $p$-torsion subgroups.
• Analyze the action of $G$ on del Pezzo surfaces using known automorphism group structures, especially for Fermat cubic and quartic del Pezzo surfaces.
• Use the $j$-invariant map $J$ from the space of 5-tuples of points on $\mathbb{P}^1$ to $\mathbb{A}^5$ to prove injectivity and properness, establishing uniqueness of conjugacy classes.
• Apply geometric and group-theoretic techniques, including blow-ups and tangent space analysis, to verify injectivity of the $j$-invariant map at generic points.

Experimental results

Research questions

• RQ1 What is the maximal rank $r$ of a $p$-elementary subgroup $G \cong (\mathbb{Z}/p)^r$ in the Cremona group $\mathrm{Cr}_k$ for $p \neq \mathrm{char}(k)$?
• RQ2 For which $p$ and $r$ is such a subgroup conjugate to the $p$-torsion subgroup of a diagonal torus in $PGL_n(k)$?
• RQ3 How many non-conjugate $p$-elementary subgroups of rank $r$ exist, particularly for $p=2$, $r=4$?
• RQ4 Why does the centralizer of a $p$-elementary subgroup in $\mathrm{Cr}_k$ not uniquely determine a maximal torus, unlike in semisimple algebraic groups?
• RQ5 What geometric structures (e.g., del Pezzo surfaces, $\mathbb{P}^1$-fibrations) support such $p$-elementary group actions?

Key findings

• For $p \geq 5$, the rank $r$ of a $p$-elementary subgroup in $\mathrm{Cr}_k$ satisfies $r \leq 2$, and if $r=2$, the subgroup is conjugate to the $p$-torsion of the diagonal torus in $PGL_3(k)$.
• For $p=3$, $r \leq 3$, and if $r=3$, the subgroup is conjugate to the $3$-torsion of the diagonal torus in $PGL_4(k)$, acting on the Fermat cubic surface $X_0^3 + \cdots + X_3^3 = 0$.
• For $p=2$, $r \leq 4$, and when $r=4$, there are two non-conjugate possibilities: one from a specific set of involutions involving rational functions, and the other from the $2$-torsion of the diagonal torus in $PGL_5(k)$ acting on a quartic del Pezzo surface in $\mathbb{P}^4$.
• The $j$-invariant map $J$ from the space of 5-tuples of points on $\mathbb{P}^1$ to $\mathbb{A}^5$ is generically injective, proven via properness and non-vanishing Jacobian determinant at generic points.
• The conjugacy class of a $2$-elementary subgroup of rank 4 is not unique, with two distinct classes arising from different geometric realizations.
• The centralizer of a $p$-elementary subgroup in $\mathrm{Cr}_k$ is strictly larger than the centralizer in a maximal torus, explaining why maximal tori are not unique, unlike in semisimple groups.