The versal deformation of small resolutions of conic bundles over $\mathbb{P}^1\times\mathbb{P}^1$ with two sections blown down

1. Problem Statement and Context

The paper addresses a specific deformation problem in complex geometry involving twistor spaces and Moishezon manifolds.

• Background: Twistor spaces are compact complex threefolds associated with self-dual Riemannian four-manifolds. For the connected sum of $n$ copies of $\mathbb{P}^2$ (denoted $n\mathbb{P}^2$), twistor spaces generally fall into two categories:
  1. LeBrun twistor spaces: Bimeromorphic to small resolutions of conic bundles over a quadric $Q = \mathbb{P}^1 \times \mathbb{P}^1$.
  2. Double solids: Small resolutions of double covers of $\mathbb{P}^3$ branched along a quartic surface.
• The Gap: While general deformation theory (Donaldson–Friedman) implies that LeBrun spaces (conic bundle type) must deform into double solids, an explicit construction of this deformation was missing. The transition is non-trivial because the two types of spaces have different geometric structures (e.g., the conic bundle has two pencils of degree 1 surfaces, while the double solid has four).
• Specific Goal: The authors aim to construct an explicit versal deformation family that connects a "Small Resolution of a Conic Bundle" (SRCB) manifold to a double solid, specifically for the case $n=3$ (relevant to $3\mathbb{P}^2$), while tracking two specific divisors (surfaces) that survive the deformation.

2. Methodology

The authors employ a combination of birational geometry, deformation theory of singularities, and explicit algebraic construction.

A. Definitions and Setup

• SRCB Manifolds: Defined as small resolutions of a threefold $Z$ obtained by blowing down two sections ($E_1, E_2$) of a conic bundle $W \subset \mathbb{P}(\mathcal{E})$ over $Q = \mathbb{P}^1 \times \mathbb{P}^1$. The bundle is defined by the equation $w_1 w_2 - \phi w_0^2 = 0$, where $\phi$ defines a discriminant curve $D$ of type $(n,n)$.
• Infinitesimal Deformation Analysis: The authors compute the cohomology of the tangent sheaf $H^i(\Theta_{\tilde{Z}})$ and the deformation space of the triple $(\tilde{Z}, \tilde{S}_1, \tilde{S}_2)$, where $\tilde{S}_i$ are strict transforms of surfaces associated with lines on $Q$. They prove that the deformation space is smooth and calculate its dimension, showing that fixing two divisors is necessary to capture the full versal deformation.

B. The Construction Strategy

The core of the paper is the construction of a family $\mathcal{Y} \to \Pi$ that interpolates between the conic bundle and the double solid via a sequence of birational transformations.

1. Extension to $\mathbb{P}^3$:

   • The discriminant curve $D$ (type $(3,3)$ for $n=3$) is extended by adding two lines (determined by a tangent plane $L$ to $Q$) to form a curve of type $(4,4)$.
   • This allows the conic bundle to be embedded in a $\mathbb{P}^2$-bundle over $\mathbb{P}^3$ rather than just over $Q$.
   • The bundle is $\mathbb{P}(\mathcal{O} \oplus \mathcal{O}(-2) \oplus \mathcal{O}(-2)) \to \mathbb{P}^3$.

2. The Interpolating Family ($\mathcal{Y}$):

   • The family is defined by two equations in the $\mathbb{P}^2$-bundle with fiber coordinates $(y_0:y_1:y_2)$:
     $$y_1 y_2 - K L y_0^2 = 0$$
     $$\alpha_2 y_1 + \alpha_1 y_2 - Q y_0 = 0$$
     Here, $Q$ is the quadric equation, $K$ is a cubic form (lifting the discriminant), and $L$ is a linear form (the tangent plane). $\alpha_1, \alpha_2$ are deformation parameters.
   • Central Fiber ($\alpha_1=\alpha_2=0$): Reducible, containing the original conic bundle components and extra sections.
   • General Fiber ($\alpha_1 \alpha_2 \neq 0$): A double solid (double cover of $\mathbb{P}^3$ branched along $Q^2 - 4\alpha_1\alpha2 KL = 0$).

3. Birational Transformations:
   To obtain a smooth family of manifolds $\tilde{\mathcal{Z}}$ that deforms the SRCB manifold, the authors perform a sequence of operations on the total space $\mathcal{Y}$:

   • Step 1 (Partial Resolution): Resolve new singularities introduced by the plane $L$ via a small resolution.
   • Step 2 (Flop): Perform a simultaneous flop of ruled surfaces ($F_1, F_2$) that appear over the divisors $\alpha_i=0$. This exchanges the "wrong" components for the "right" ones.
   • Step 3 (Contraction): Blow down specific components to recover the structure of a double solid or the original conic bundle depending on the parameters.
   • Step 4 (Final Resolution): Resolve remaining singularities to ensure the fibers are smooth manifolds.

3. Key Contributions and Results

A. Explicit Construction of the Versal Family

The authors provide explicit equations for a family $\tilde{\mathcal{Z}} \to \Pi$ where:

• The fiber over $\alpha_1 = \alpha_2 = 0$ is an SRCB manifold.
• The fiber over $\alpha_1 \alpha_2 \neq 0$ is a double solid.
• The family is versal for deformations of the triple $(\tilde{Z}, \tilde{S}_1, \tilde{S}_2)$.

B. Theorem on Versality

Theorem: The constructed family $\tilde{\mathcal{Z}} \to \Pi$ is versal for deformations of triples $(\tilde{Z}, \tilde{S}_1, \tilde{S}_2)$ near the locus of SRCB manifolds.

• Proof Strategy: They compute the tangent space of the deformation functor $T^1_f$ and show the Kodaira-Spencer map is surjective. This involves analyzing the kernel of the map $H^1(\Theta_{\tilde{Z}}) \to H^1(N_{\tilde{S}_1}) \oplus H^1(N_{\tilde{S}_2})$ and showing the parameters $\alpha_1, \alpha_2$ span the necessary directions.

C. Analysis of Intermediate Fibers

The paper analyzes fibers where only one parameter is zero (e.g., $\alpha_1 \neq 0, \alpha_2 = 0$). These fibers represent "halfway" states between conic bundles and double solids. They are shown to be modifications of $\mathbb{P}^3$ involving blow-ups of lines and curves, containing one surviving pencil of surfaces. These manifolds do not admit a real structure compatible with the twistor definition, explaining why they are not standard twistor spaces.

D. Applications to Twistor Theory

• Honda's Result: The construction provides a new proof of a result by Honda regarding the existence of degenerate double solids (with higher singularities) as twistor spaces.
• Real Structure: For LeBrun twistor spaces (where the discriminant splits into three linear forms), the authors show how the real structure is preserved in the deformation, confirming that small real deformations of LeBrun spaces yield double solid twistor spaces.
• Symmetry and Flops: In the case of torus actions, the paper demonstrates that interchanging the rulings of the quadric corresponds to a flop of all exceptional curves in the double solid, yielding two distinct small resolutions that admit a twistor structure.

E. Specific Examples

The authors apply their general construction to specific cases:

1. Honda's Deformation (Torus Action): Explicitly describes the deformation for spaces with $C^*$ symmetry, deriving the dependence of the deformation parameter $s$ on $\alpha_1 \alpha_2$ and identifying the resulting curve in the parameter space (a rational sextic with cusps).
2. Quartics with 14 Double Points: Constructs a family where the general fiber is a double solid branched along a quartic with 14 nodes.
3. Kummer Surfaces: Relates the deformation to the moduli space of Kummer surfaces (Igusa quartic), showing how the family arises from a common resolution of the Segre cubic and the Igusa quartic.

4. Significance

• Bridging Two Worlds: The paper successfully bridges the gap between two major classes of twistor spaces (conic bundles and double solids) by providing an explicit algebraic mechanism for their deformation.
• Resolution of Singularities: It offers a sophisticated method for handling simultaneous small resolutions and flops in families, demonstrating how to navigate the "moduli space" of singular threefolds.
• Twistor Geometry: It clarifies the structure of the moduli space of twistor spaces over $3\mathbb{P}^2$, showing that the transition from LeBrun spaces to double solids is not just a theoretical existence result but a concrete geometric process involving specific birational transformations.
• Generalizability: While focused on $n=3$, the methodology (infinitesimal deformation analysis and the construction of the family $\mathcal{Y}$) is presented in a way that suggests potential extensions to $n \ge 4$, addressing open questions about the deformation of generalized LeBrun spaces.

In summary, Kreussler and Stevens provide a rigorous, constructive proof of the deformation relationship between conic bundle twistor spaces and double solid twistor spaces, utilizing advanced techniques in birational geometry to resolve singularities and track the evolution of specific divisors.