Remarks on the group of birational selfmaps of a conic fibration

1. Problem Statement and Context

The paper investigates the structure of the group of birational self-maps, denoted $\text{Bir}(X)$, for projective varieties $X$ that are birational to conic fibrations (varieties fibered over a base $Y$ with general fibers being rational curves).

• Background: While the group $\text{Bir}(\mathbb{P}^n)$ is well-studied, the structure of $\text{Bir}(X)$ for non-rational rationally connected varieties is less understood. A recent result suggests that if $\text{Bir}(X) \cong \text{Bir}(\mathbb{P}^n)$, then $X$ must be rational.
• The Gap: It was previously known that $\text{Bir}(X)$ for conic bundles is uncountable (due to its isomorphism to an orthogonal group over the function field $C(Y)$). However, the specific algebraic structure and the nature of its generators were not fully characterized in terms of explicit homomorphisms to large abelian groups.
• Goal: The author aims to prove that the group $\text{Bir}(X/Y)$ (birational maps preserving the fibration) admits a surjective morphism to an uncountable direct sum of copies of $\mathbb{Z}/2\mathbb{Z}$, thereby establishing a rich, non-discrete structure even for non-rational varieties.

2. Methodology

The paper employs a combination of Birational Geometry (MMP), Algebraic Group Theory (Orthogonal Groups), and Topological methods on birational groups.

A. Geometric Construction of Involutions

The core geometric tool is the construction of Type II links (specifically birational involutions) on conic bundles.

• Setup: Let $f: X \to Y$ be a conic bundle. The author considers rational sections $s: Y \dashrightarrow \mathbb{P}(E)$ (where $X \subset \mathbb{P}(E)$).
• Involution $\tau_s$: For a section $s$ not contained in $X$, an involution $\tau_s$ is defined fiber-wise by projecting from the point $s(y)$ on the conic fiber.
• Indeterminacy Locus: The indeterminacy locus of $\tau_s$ is determined by the intersection of the section with the conic bundle. The author proves that by choosing sections with sufficiently high degree, one can generate uncountably many distinct indeterminacy loci $\Gamma$.

B. Algebraic Interpretation via Spinor Norm

The paper bridges the geometric construction with the theory of Orthogonal Groups.

• Isomorphism: $\text{Bir}(X/Y)$ is isomorphic to the automorphism group of the generic fiber $X_\eta$, which is a smooth quadric over the function field $F = C(Y)$. Thus, $\text{Bir}(X/Y) \cong \text{Aut}(X_\eta) \subset O(F^3)$.
• Spinor Norm: The author utilizes the spinor norm homomorphism $\theta: O(V) \to F^*/(F^*)^2$.
• Factorization: The main homomorphism $\Phi$ (previously defined in [BLZ21]) is shown to factor through the spinor norm and a divisor map:
  $$\Phi = \xi \circ \text{div} \circ \theta$$
  where $\text{div}$ maps the class of a rational function to its divisor modulo 2.

C. Topological Analysis

The paper analyzes the topology of $\text{Bir}(X)$ using the inductive limit topology defined by Hilbert schemes (as studied in [RUvS25b]).

• Bounded Degree: A subset of $\text{Bir}(X)$ has "bounded degree" if it is contained in a finite union of Hilbert schemes corresponding to bounded Hilbert polynomials.
• Strategy: The author proves that the generators of $\text{Bir}(X)$ cannot be bounded in degree by showing that the required degrees for the defining sections grow indefinitely.

3. Key Contributions and Results

Theorem 1.1: Uncountable Abelian Quotient

Statement: Let $f: X \to Y$ be a fibration with rational curve fibers. There exists an uncountable set $I$ and a surjective homomorphism:
$$\Phi: \text{Bir}(X/Y) \to \bigoplus_{I} \mathbb{Z}/2\mathbb{Z}$$
Significance: This proves that the group of birational self-maps is not just uncountable, but possesses a massive abelian quotient structure. This distinguishes it sharply from groups that might be "small" or discrete.

Theorem 1.2: Factorization via Spinor Norm

Statement: The homomorphism $\Phi$ factors as $\xi \circ \text{div} \circ \theta$, where:

1. $\theta$ is the spinor norm mapping to $C(Y)^*/(C(Y)^*)^2$.
2. $\text{div}$ maps the class of a function to the sum of its prime divisors modulo 2.
3. $\xi$ is a projection to the uncountable sum.
   Significance: This provides a concrete algebraic interpretation of the geometric involutions constructed in [BLZ21], linking the geometry of the indeterminacy locus directly to the arithmetic of the function field via the spinor norm.

Corollary 1.3: Non-Rationality and Uncountability

Statement: If $X$ is a 3-dimensional rationally connected, non-rational variety birational to a conic bundle, then $\text{Bir}(X)$ is uncountable, yet the connected component of the automorphism group $\text{Aut}^\circ(Y)$ is trivial for any $Y$ birational to $X$.
Significance: This resolves a potential conjecture that non-rational varieties might have "discrete" birational groups. It shows that even when the continuous automorphism group is trivial, the birational group can be vast.

Corollary 1.4: Unbounded Generators

Statement: If $\dim(Y) \geq 2$, there is no set of generators for $\text{Bir}(X)$ of bounded degree.
Significance: This is a topological result. It implies that the group $\text{Bir}(X)$ is not finitely generated in any bounded sense; one requires algebraic families of generators of arbitrarily high degree to generate the group. This contrasts with the structure of $\text{Bir}(\mathbb{P}^2)$ or $\text{Bir}(\mathbb{P}^n)$ in certain contexts.

4. Technical Lemmas Supporting the Proofs

• Lemma 3.1 (Isotriviality): Establishes that fibrations with general type fibers that are birational to each other are essentially products modulo a finite group action. This is used to rule out certain birational equivalences between fibers.
• Lemma 3.2 (Non-birationality of Intersections): Proves that general complete intersections of codimension 2 inside a conic bundle (with sufficiently high degree) are not birational to each other. This is the crucial step to ensure the set $I$ in Theorem 1.1 is uncountable. If all such intersections were birational, the quotient would be countable.
• Lemma 3.3 (Embedding): Shows that any conic fibration can be birationally transformed into a subvariety of a projective bundle $\mathbb{P}(\mathcal{O} \oplus \mathcal{O}(a) \oplus \mathcal{O}(b))$, allowing the use of explicit equations for the spinor norm calculation.

5. Significance and Impact

1. Structural Insight: The paper significantly advances the understanding of the group structure of birational maps for non-rational varieties. It moves beyond the simple fact of "uncountability" to describe the specific algebraic structure (a large direct sum of $\mathbb{Z}/2\mathbb{Z}$).
2. Connection to MMP: It connects the Minimal Model Program (MMP) outcomes (conic bundles) with deep group-theoretic properties, showing that the "birational complexity" of these varieties is immense.
3. Topological Constraints: The result on unbounded degrees (Corollary 1.4) imposes strong constraints on the topology of the space of birational maps, suggesting that the "space" of birational maps for conic bundles is much more complex than that of rational varieties.
4. Refinement of Previous Work: It refines the results of Blanc, Lamy, and Zimmermann ([BLZ21]) by providing the explicit factorization through the spinor norm and proving the uncountability of the index set $I$ rigorously using intersection theory.

In summary, Floris demonstrates that the group of birational self-maps of a conic fibration is structurally rich, admitting a massive abelian quotient derived from the spinor norm, and requires generators of arbitrarily high degree, distinguishing it fundamentally from the birational groups of rational varieties.