Conic bundles and Clifford algebras

TL;DR

This paper explores the deep connections between quadratic forms, conic bundles, and quaternion orders using Clifford algebras and Brauer-Severi varieties, providing new insights into their classification and invariants.

Contribution

It establishes natural bijections between quadratic forms, conic bundles, and quaternion orders, and relates their invariants, especially on surfaces, including minimal del Pezzo quaternion orders.

Findings

Derived formulas for the second Chern class in terms of invariants.

Identified conic bundles corresponding to minimal del Pezzo quaternion orders.

Discussed moduli problems related to these geometric objects.

Abstract

We discuss natural connections between three objects: quadratic forms with values in line bundles, conic bundles and quaternion orders. We use the even Clifford algebra, and the Brauer-Severi Variety, and other constructions to give natural bijections between these objects under appropriate hypothesis. We then restrict to a surface base and we express the second Chern class of the order in terms $K^3$ and other invariants of the corresponding conic bundle. We find the conic bundles corresponding to minimal del Pezzo quaterion orders and we discuss problems concerning their moduli.