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 studies the deformation theory of small resolutions of conic bundles over 11, providing explicit descriptions of their versal deformations and connecting them to twistor spaces over 32.

Contribution

It offers a detailed analysis of the versal deformation space of small resolutions of conic bundles over 11, extending understanding of their deformation behavior.

Findings

Explicit description of versal deformations of small resolutions.

Connection between conic bundle deformations and twistor spaces.

Demonstration of how these deformations relate to double solids.

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.