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

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.Comment: 23 pages; Macaulay 2 code used for verification of parts of the paper available at http://www.math.uni-hamburg.de/home/bothmer/m2.html and at the end of the TeX file; v2: in section 4, we now included a proof of the main theorem that works for all n (unconditional on the parity) that was communicated to us by Zhi Jiang, Zhiyu Tian, and Letao Zhang. Several other minor expository improvement

Degenerations of Gushel-Mukai fourfolds, with a view towards irrationality proofs

We study a certain class of degenerations of Gushel-Mukai fourfolds as conic bundles, which we call tame degenerations and which are natural if one wants to prove that very general Gushel-Mukai fourfolds are irrational using the degeneration method due to Voisin, Colliot-Th\'{e}l\`{e}ne-Pirutka, Totaro et al. However, we prove that no such tame degenerations do exist.Comment: 25 page

Birationally isotrivial fiber spaces

We prove that a family of varieties is birationally isotrivial if all the fibers are birational to each other.Comment: 10 pages; v2: thanks to helpful suggestions of the referee, the main result is now independent of the Eisenbud-Goto conjecture; also we removed the erroneous Prop. 3.4 of v1 which was not needed for anything else in the pape