On stable rationality of some conic bundles and moduli spaces of Prym curves

TL;DR

This paper demonstrates that certain hypersurfaces in product spaces are not stably rational by employing Chow-theoretic methods, and also explores degenerations within Prym moduli spaces using elementary techniques.

Contribution

It introduces an elementary approach to proving non-stable rationality of specific hypersurfaces and analyzes degenerations in Prym moduli spaces without advanced deformation theory.

Findings

Very general hypersurfaces of bidegree (2, n) in P^2 x P^2 are not stably rational for n ≥ 2.

Provides explicit degeneration analysis of Prym curves within their moduli space.

Employs Koszul complexes instead of deformation theory for these results.

Abstract

We prove that a very general hypersurface of bidegree (2, n) in P^2 x P^2 for n bigger than or equal to 2 is not stably rational, using Voisin's method of integral Chow-theoretic decompositions of the diagonal and their preservation under mild degenerations. At the same time, we also analyse possible ways to degenerate Prym curves, and the way how various loci inside the moduli space of stable Prym curves are nested. No deformation theory of stacks or sheaves of Azumaya algebras like in recent work of Hasset-Kresch-Tschinkel is used, rather we employ a more elementary and explicit approach via Koszul complexes, which is enough to treat this special case.