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

Bernd Kreussler, Jan Stevens

TL;DR

This paper constructs and analyzes a versal deformation for small resolutions of conic bundles over Q = P^1 × P^1 with two sections blown down (SRCB manifolds), focusing on the case n = 3 and linking SRCB deformations to double solids. The authors develop a detailed birational deformation framework (Y → Π, Y^−, Y^+, Z, and ilde Z) to interpolate between conic-bundle twistor spaces and double solids, and they compute infinitesimal deformation spaces to establish smooth versality for the triple (tilde Z, tilde S1, tilde S2). They provide explicit constructions and stratifications of the discriminant data, show that the deformation spaces are controlled by delta-constant degenerations of the discriminant curve, and give concrete examples including LeBrun twistor spaces, quartics with 14 double points, and Kummer surfaces. The work extends known twistor-space results (including Honda’s torus-symmetric cases) and supplies a geometric path showing how Moishezon twistor spaces of conic-bundle type deform into small resolutions of double solids, with implications for the moduli of special complex threefolds.

Abstract

Twistor spaces are certain compact complex threefolds with an additional real fibre bundle structure. We focus here on twistor spaces over $3\mathbb{C}\mathbb{P}^2$. Such spaces are either small resolutions of double solids or they can be described as modifications of conic bundles. The last type is the more special one: they deform into double solids. We give an explicit description of this deformation, in a more general context.

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

TL;DR

This paper constructs and analyzes a versal deformation for small resolutions of conic bundles over Q = P^1 × P^1 with two sections blown down (SRCB manifolds), focusing on the case n = 3 and linking SRCB deformations to double solids. The authors develop a detailed birational deformation framework (Y → Π, Y^−, Y^+, Z, and ilde Z) to interpolate between conic-bundle twistor spaces and double solids, and they compute infinitesimal deformation spaces to establish smooth versality for the triple (tilde Z, tilde S1, tilde S2). They provide explicit constructions and stratifications of the discriminant data, show that the deformation spaces are controlled by delta-constant degenerations of the discriminant curve, and give concrete examples including LeBrun twistor spaces, quartics with 14 double points, and Kummer surfaces. The work extends known twistor-space results (including Honda’s torus-symmetric cases) and supplies a geometric path showing how Moishezon twistor spaces of conic-bundle type deform into small resolutions of double solids, with implications for the moduli of special complex threefolds.

Abstract

Twistor spaces are certain compact complex threefolds with an additional real fibre bundle structure. We focus here on twistor spaces over . Such spaces are either small resolutions of double solids or they can be described as modifications of conic bundles. The last type is the more special one: they deform into double solids. We give an explicit description of this deformation, in a more general context.
Paper Structure (15 sections, 16 theorems, 106 equations, 5 figures)

This paper contains 15 sections, 16 theorems, 106 equations, 5 figures.

Table of Contents

  1. SRCB manifolds
  2. Infinitesimal deformations
  3. A versal family for $n=3$
  4. The family ${\mathcal{Y}} \to\Pi$
  5. The family $\widetilde{{\mathcal{Z}}} \to\Pi$
  6. ${\mathcal{Y}}^-\longrightarrow{\mathcal{Y}}$
  7. ${\mathcal{Y}}^{-} \dashrightarrow {\mathcal{Y}}^{+}$
  8. ${\mathcal{Y}}^+\longrightarrow {\mathcal{Z}}$
  9. $\widetilde{{\mathcal{Z}}}\longrightarrow {\mathcal{Z}}$
  10. The main theorem
  11. The fibres over $\Delta_i$
  12. Examples
  13. Twistor spaces
  14. Quartics with 14 double points
  15. Kummer surfaces

Key Result

Theorem 1

The fibres of the so-constructed family $\widetilde{{\mathcal{Z}}}\rightarrow\Pi$ over $\alpha_{1}=\alpha_{2}=0$ are SRCB manifolds, and the deformation $\widetilde{{\mathcal{Z}}}\rightarrow\Pi$ is for small $\alpha_{1},\alpha_{2}$ versal for deformations of triples $(\widetilde{Z},\widetilde{S}_1,\

Figures (5)

  • Figure 1: Factorisation of the map $Y^+ \dashrightarrow Y$.
  • Figure 2: Factorisation in case $P$ lies on the discriminant curve.
  • Figure 3: The dependence of $s$ on $\alpha_{1}\alpha_{2}$ with a far away component at another scale.
  • Figure 4: Enlarged view around the origin.
  • Figure 5: Discriminant curve for a 14-nodal conic bundle.

Theorems & Definitions (40)

  • Theorem
  • Remark 1.1
  • Theorem 1.2: Castelnuovo--Moishezon--Nakano criterion
  • Definition
  • Example 1.3
  • Remark 1.4
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 30 more