On the unirationality of conic bundles with discriminant of degree eight

TL;DR

This paper investigates the unirationality of conic bundle surfaces over arbitrary fields with discriminant degree eight, providing explicit constructions and analyzing their properties and deformations.

Contribution

It classifies conic bundles with discriminant degree eight into four families, constructs explicit rational multisections, and studies their unirationality and deformation behavior.

Findings

Constructed explicit rational multisections for each family.

Identified Zariski dense loci of non-rational yet unirational conic bundles.

Proved Zariski openness of unirational loci in one family.

Abstract

We study the unirationality of surface conic bundles $\pi\colon S\to\mathbb P^1$ over an arbitrary field $k$ with discriminant degree $d_S=8$, the first case beyond the del Pezzo range. We divide these surfaces in four families and produce explicit rational multisections via tangent constructions and Cremona transformations. Over $C_1$ fields we obtain Zariski dense loci of minimal, hence non $k$-rational, yet $k$-unirational conic bundles in each family; for one of the types we prove that the dense unirational locus is indeed Zariski open. Finally, we investigate the deformation theory of these conic bundles and how their unirationality behaves under specialization.