Relative Chow-Kunneth decompositions for conic bundles and Prym varieties

TL;DR

This paper constructs a refined decomposition of conic bundle motives, linking the Prym variety to the intermediate Jacobian, enhancing Beauville's theorem on their relationship.

Contribution

It introduces a relative Chow-Kunneth decomposition for conic bundles that explicitly incorporates Prym varieties, refining existing theorems.

Findings

Established a relative Chow-Kunneth decomposition for conic bundles.

Connected the middle projector to the Prym variety of a double covering.

Refined Beauville's theorem up to an isogeny.

Abstract

We construct a relative Chow-Kunneth decomposition for a conic bundle over a surface such that the middle projector gives the Prym variety of the associated double covering of the discriminant of the conic bundle. This gives a refinement (up to an isogeny) of Beauville's theorem on the relation between the intermediate Jacobian of the conic bundle and the Prym variety of the double covering.