Automorphism groups of $\mathbb{P}^1$-bundles over ruled surfaces

This paper classifies all pairs consisting of a $\mathbb{P}^1$-bundle over a non-rational ruled surface and its relatively maximal connected automorphism group over an algebraically closed field of characteristic zero.

The Gist

Imagine you are an architect trying to understand the most stable, "maximal" structures you can build in a very specific, magical universe. This universe is governed by the rules of algebraic geometry, but let's translate it into something more relatable: a world of infinite, twisting towers and bridges.

The Setting: The City of Curves and Towers

In this paper, the author, Pascal Fong, is studying a specific type of building called a $\mathbb{P}^1$-bundle.

• The Ground ($C$): Imagine a smooth, closed loop, like a rubber band or a circle. In math terms, this is a "curve of genus $g \ge 1$." If it's a simple circle, it's an "elliptic curve" (like a donut shape). If it has more holes, it's a more complex loop.
• The Base Floor ($S$): On top of this rubber band, we build a "ruled surface." Think of this as a long hallway where every single point on the rubber band has a small, straight line (like a ruler) attached to it, pointing upwards. This hallway is our surface $S$.
• The Tower ($X$): Now, we build a second layer on top of that hallway. At every point on the hallway, instead of just a line, we attach a whole circle (a $\mathbb{P}^1$). So, our final building $X$ is a 3D structure: a rubber band $\to$ a hallway $\to$ a tower of circles.

The Goal: Finding the "Maximal" Symmetry Groups

Every building has symmetries. You can rotate it, slide it, or flip it, and it might look the same. In math, these symmetries form a group.

• The Problem: There are infinitely many ways to build these towers. Some are rigid, some are wobbly. Fong wants to find the "Maximal Connected Algebraic Subgroups."
• The Translation: He is looking for the biggest, most robust "symmetry clubs" possible. He wants to know: Which of these towers has the largest possible group of symmetries that can't be made any bigger without breaking the rules of the universe?

He calls these "Relatively Maximal." It's like finding the strongest possible team of dancers for a specific dance floor. If you try to add one more dancer, the choreography falls apart.

The Strategy: The "Minimal Model" Game

To find these strongest teams, Fong uses a strategy called the Minimal Model Program (MMP).

• The Analogy: Imagine you have a messy, overgrown garden (your building). You want to find its "essence." You start pruning. You cut away unnecessary branches (singularities) and reshape the garden until you get a clean, standard shape (a "Mori fiber space").
• The Process: Fong takes any weird, complex tower and "prunes" it using birational maps (which are like folding and unfolding paper without tearing it). He asks: Can I fold this tower into a simpler, standard shape that still keeps all its symmetry?
• The Sarkisov Program: This is the specific rulebook for how to fold and unfold these towers. It's like a game of Tetris where you can only move pieces in specific ways to see if you can reach a "winning" configuration (a maximal symmetry group).

The Big Discovery: The "Unbounded" Surprise

Here is the twist that makes this paper famous.

• The Old Belief: In simpler universes (like the 2D plane), mathematicians thought every symmetry group was contained inside a "maximal" one. Like every small club is part of a bigger, ultimate club.
• The New Reality: Fong discovers that in this 3D world of towers over loops, this is not always true.
  • There are "unbounded" symmetry groups. These are like clubs that keep growing forever. No matter how big you make them, you can always add more members, but you never reach a "final boss" or a "maximal" version. They just keep expanding.
  • However, Fong successfully classifies the ones that do stop growing. He finds the "Final Bosses" of symmetry.

The Classification: The "Menu" of Maximal Towers

Fong creates a menu of all the possible "Final Boss" towers. They fall into two main categories based on the shape of the ground loop ($C$):

1. If the ground is a complex loop (Genus $\ge 2$):

• There is only one type of maximal tower: The Trivial Tower.
• Analogy: Imagine a perfect, boring, straight skyscraper where every floor is identical and perfectly aligned. It's the product of the loop, a hallway, and a circle ($C \times \mathbb{P}^1 \times \mathbb{P}^1$). It's the only one that can't be improved.

2. If the ground is a simple loop (Genus = 1, an Elliptic Curve):

• This is where it gets spicy! Because the loop is a simple circle, there are more ways to twist the tower.
• Fong lists eight different types of maximal towers.
  • Some are just products (like stacking identical blocks).
  • Some are "indecomposable," meaning they are twisted so tightly they can't be pulled apart into simple layers.
  • Some involve "Atiyah's ruled surfaces" (special, twisted hallways named after a famous mathematician).
  • Some involve "decomposable" bundles where the tower splits into two distinct parts.

The "Stiffness" Concept

Fong also introduces a concept called "Stiffness" (and "Superstiffness").

• Stiff: Imagine a building that is so unique that if you try to fold it or reshape it (even slightly), you lose its symmetry. It's rigid.
• Superstiff: This is the ultimate rigidity. The building is so unique that the only way to reshape it into another building with the same symmetry is to do absolutely nothing (it's an isomorphism). It's a "one-of-a-kind" masterpiece.
• Not Stiff: Some towers are flexible. You can fold them into different shapes, and they will still have the same symmetry group. They are like origami that can be folded into a crane or a boat, but the "folding club" remains the same.

Why Does This Matter?

This paper is like a catalog of the most stable structures in a specific mathematical universe.

• It solves a puzzle that started in the 1920s (Enriques' classification) but got stuck in higher dimensions.
• It reveals that in 3D, things are wilder than we thought (the "unbounded" groups).
• It provides a complete map for anyone trying to navigate the symmetries of these complex towers.

In a nutshell: Pascal Fong took a chaotic, infinite landscape of 3D towers, used a set of folding rules (Sarkisov links) to prune them down, and found the specific, unshakeable "King Towers" that hold the most symmetry possible. He showed us that while some towers can grow forever, others are the absolute limit of stability, and he gave us the blueprints for all of them.

Technical Summary

Here is a detailed technical summary of the paper "Automorphism Groups of P1-Bundles over Ruled Surfaces" by Pascal Fong.

1. Problem Statement

The paper addresses the classification of maximal connected algebraic subgroups of the Cremona group $\text{Bir}(X)$, where $X$ is a specific type of threefold. Specifically, the author investigates pairs $(X, \pi)$ where:

• $C$ is a smooth projective curve of genus $g \ge 1$.
• $S$ is a geometrically ruled surface over $C$ (a $\mathbb{P}^1$-bundle over $C$).
• $\pi: X \to S$ is a $\mathbb{P}^1$-bundle over $S$ (making $X$ a threefold fibered over $C$ with fibers isomorphic to $\mathbb{P}^1 \times \mathbb{P}^1$ or Hirzebruch surfaces).
• The goal is to classify those $X$ for which the connected component of the automorphism group, $\text{Aut}^\circ(X)$, is relatively maximal.

Key Definitions:

• Relatively Maximal: $\text{Aut}^\circ(X)$ is relatively maximal if, for every $\text{Aut}^\circ(X)$-equivariant square birational map $\phi: X \dashrightarrow X'$ (where $X'$ is another $\mathbb{P}^1$-bundle over a ruled surface), the conjugated group $\phi \text{Aut}^\circ(X) \phi^{-1}$ equals $\text{Aut}^\circ(X')$.
• Stiffness: A pair $(X, \pi)$ is stiff if any such equivariant square birational map is an isomorphism. It is superstiff if the map is an isomorphism on both the total space and the base.
• Context: Previous work (Enriques, Umemura, Blanc-Fanelli-Terpereau) classified maximal subgroups for rational surfaces ($\mathbb{P}^2, \mathbb{P}^3$) and ruled surfaces over curves ($C \times \mathbb{P}^1$). A key phenomenon discovered in [Fon21] is the existence of unbounded connected algebraic subgroups in $\text{Bir}(C \times \mathbb{P}^1)$ (subgroups not contained in any maximal one). This paper extends the classification to the next dimension: $\mathbb{P}^1$-bundles over ruled surfaces.

2. Methodology

The author employs a strategy combining the Minimal Model Program (MMP) and the Sarkisov Program, adapted to the equivariant setting for connected algebraic groups.

1. Reduction to F-bundles:

   • Using the regularization theorem and MMP, any connected algebraic subgroup $G \subset \text{Bir}(X)$ can be conjugated to a subgroup of $\text{Aut}^\circ(X)$ where $X$ is a conic bundle over a ruled surface.
   • The author proves that if $\text{Aut}^\circ(X)$ is relatively maximal, the morphism $\tau\pi: X \to C$ must be an $\mathbb{F}_b$-bundle (a bundle with fibers isomorphic to the $b$-th Hirzebruch surface $\mathbb{F}_b$) over $C$.
   • Jumping Fibers: The paper eliminates "jumping fibers" (fibers isomorphic to $\mathbb{F}_{b'}$ with $b' > b$) via equivariant birational modifications, ensuring the generic fiber is $\mathbb{F}_b$.

2. Case Analysis based on $b$:

   • Case $b=0$: The bundle is an $\mathbb{F}_0$-bundle, which corresponds to a fiber product of two ruled surfaces over $C$ ($X \cong S_1 \times_C S_2$). The automorphism group is analyzed as a fiber product of the automorphism groups of the factors.
   • Case $b > 0$: The author utilizes the Sarkisov Program to analyze equivariant links.
     • If $b \ge 2$, no Sarkisov links of type III or IV exist that preserve terminal singularities.
     • If $b=1$, a link to a $\mathbb{P}^2$-bundle exists.
     • The author constructs explicit invariants $(S, b, D)$ for the bundle, where $S$ is the base ruled surface, $b$ is the Hirzebruch index, and $D$ is a divisor class on $C$.
     • By studying the action of $\text{Aut}^\circ(X)$ on these invariants and using Blanchard's Lemma (which relates $\text{Aut}^\circ(X)$ to $\text{Aut}^\circ(S)$), the author determines when the group is maximal.

3. Specific Surfaces:
   The analysis is broken down by the type of the base ruled surface $S$:

   • $S = C \times \mathbb{P}^1$ (Trivial ruled surface).
   • $S = A_0$ and $S = A_1$ (Atiyah's indecomposable ruled surfaces over an elliptic curve).
   • $S = \mathbb{P}(\mathcal{O}_C \oplus \mathcal{L})$ (Decomposable ruled surfaces with non-trivial degree 0 line bundle).

3. Key Contributions and Results

Theorem A: Classification of Relatively Maximal Pairs

The paper provides a complete list of pairs $(X, \pi)$ where $\text{Aut}^\circ(X)$ is relatively maximal, distinguishing between genus $g=1$ (elliptic curve) and $g \ge 2$.

• Case $g \ge 2$:

  • The only relatively maximal case is the trivial bundle $X = C \times \mathbb{P}^1 \times \mathbb{P}^1$ over $S = C \times \mathbb{P}^1$.
  • $\text{Aut}^\circ(X) \cong \text{PGL}_2(k) \times \text{PGL}_2(k)$.

• Case $g = 1$ (Elliptic Curve):
  The classification is much richer and includes:

  1. Fiber Products: $X = S' \times_C S''$ where $S', S''$ are ruled surfaces from the list of maximal subgroups of $\text{Bir}(C \times \mathbb{P}^1)$ (i.e., $C \times \mathbb{P}^1$, $A_0$, $A_1$, or $\mathbb{P}(\mathcal{O} \oplus \mathcal{L})$).
     • Examples include $A_0 \times_C A_1$, $A_1 \times_C A_1$, and products involving $\mathbb{P}(\mathcal{O} \oplus \mathcal{L})$.
  2. Non-trivial Bundles ($b > 0$):
     • Decomposable bundles of the form $X \cong \mathbb{P}(\mathcal{O}_S \oplus \mathcal{O}_S(b\sigma + \tau^*(D)))$.
     • Specific conditions on the divisor $D$ (e.g., non-trivial, not 2-torsion, or infinite order) are required for maximality.
     • Notably, there are cases where $X$ is indecomposable (e.g., specific bundles over $A_1$ or $A_0$) that still yield relatively maximal groups.

Theorem C: Stiffness and Conjugacy Classes

The paper determines which of these maximal groups are stiff (unique in their conjugacy class) and describes the square birational maps connecting non-stiff models.

• Superstiff Cases: Many fiber products (like $C \times \mathbb{P}^1 \times \mathbb{P}^1$, $A_0 \times \mathbb{P}^1$, $A_1 \times \mathbb{P}^1$, $A_1 \times_C A_1$) are superstiff.
• Non-Stiff Cases: For bundles involving $\mathbb{P}(\mathcal{O} \oplus \mathcal{L})$ or specific indecomposable bundles, there exist infinite families of non-isomorphic $\mathbb{P}^1$-bundles $Y$ such that $\text{Aut}^\circ(X)$ and $\text{Aut}^\circ(Y)$ are conjugate via square birational maps.
  • The paper explicitly describes these families using parameters $b$ and $n$ (e.g., $Y \cong \mathbb{P}(\mathcal{O}_{A_1} \oplus \mathcal{O}_{A_1}(4n\sigma + \tau^*(D - nD_\sigma)))$).

Proposition B: Structure of Automorphism Groups

For each classified case, the paper computes:

• The dimension of $\text{Aut}^\circ(X)$.
• The kernel of the morphism $\pi_*: \text{Aut}^\circ(X) \to \text{Aut}^\circ(S)$.
• The orbit structure of the action on $X$ (e.g., number of orbits, their dimensions, and whether they are transitive).
  • Example: For $X = A_0 \times_C A_1$, the group has dimension 2 and acts with orbits of dimension 1 and 2.

Corollary D: Maximality in $\text{Bir}(X)$

The paper distinguishes between "relatively maximal" (maximal within the category of $\mathbb{P}^1$-bundles over ruled surfaces) and "maximal" in the full Cremona group $\text{Bir}(X)$.

• For $g \ge 2$, the unique relatively maximal group is also maximal in $\text{Bir}(X)$.
• For $g=1$, some relatively maximal groups are not maximal in $\text{Bir}(X)$ because they can be linked to Mori del Pezzo fibrations via equivariant birational maps (specifically in cases involving $A_0 \times_C A_1$).

4. Significance

1. Extension of Classification: This work significantly extends the classification of maximal algebraic subgroups of Cremona groups from rational surfaces and $C \times \mathbb{P}^1$ to the next level of complexity: threefolds fibered over curves.
2. Handling Unbounded Subgroups: It addresses the phenomenon of unbounded connected algebraic subgroups (subgroups not contained in any maximal one) by identifying the "relatively maximal" ones that serve as the boundaries of these unbounded families.
3. Equivariant Sarkisov Program: The paper provides a concrete application of the equivariant Sarkisov program in dimension 3, explicitly calculating the links (blow-ups and contractions) that relate different models of the same birational class.
4. Detailed Orbit Analysis: Unlike previous classifications that focused primarily on group structure, this paper provides a granular analysis of the orbit decomposition of the group action, which is crucial for understanding the geometry of the moduli spaces involved.
5. Foundation for Future Work: The results lay the groundwork for the complete classification of maximal subgroups in $\text{Bir}(C \times \mathbb{P}^2)$ by isolating the $\mathbb{P}^1$-bundle cases and pointing toward the remaining cases (Mori del Pezzo fibrations) for future study.

In summary, Pascal Fong's paper provides a rigorous and exhaustive classification of the symmetry groups of a specific class of threefolds, revealing a rich structure of conjugacy classes, stiffness properties, and geometric invariants that depend heavily on the arithmetic of the base curve (genus 1 vs. genus $\ge 2$) and the torsion properties of line bundles.