Unirational threefolds with no universal codimension $$2$$2 cycle

By Claire Voisin. 2013

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We prove that the general quartic double solid with $$k\le 7$$k≤7 nodes does not admit a Chow theoretic decomposition of the diagonal, (or equivalently has a nontrivial universal $$\mathrm{CH}_0$$CH0 group,) and the same holds if we replace in this statement “Chow theoretic” by “cohomological”. In particular, it is not stably rational. We also deduce that the general quartic double solid with seven nodes does not admit a universal codimension $$2$$2 cycle parameterized by its intermediate Jacobian, and even does not admit a parametrization with rationally connected fibers of its Jacobian by a family of $$1$$1-cycles. This finally implies that its third unramified cohomology group is not universally trivial.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/313b4f671351a55860dbe885054708cc65f307c6

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In Geometry and Topology

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Citations in the corpus

Measures of irrationality for hypersurfaces of large degree (2015, methodology): “In this setting, Ein and Voisin prove that if t ∈ T is a general point, then Ωv+s S ⊗ ( (pr2 ◦ f)∗OPn+1(2n+ 2− d− s) ) ∣ ∣ ∣ St is generically generated by its global sections (cf [32, Theorem 1.4]), where St = π −1(t) is the fibre of π.”
The motivic nearby fiber and degeneration of stable rationality (2017, methodology): “ot stably rational because it has nonvanishing unramiﬁed Brauer group. It follows that very general smooth quartic double solids are not stably rational either. This has already been proved by Voisin [Vo15] using the degeneration method for Chow groups of zero-cycles and the same degeneration X 0. A”
Stable rationality of quadric surface bundles over surfaces (2016, methodology): “The ﬁrst author was partially supported through NSF grant 1551514. 2. Generalities We recall implications of the “integral decomposition of the diagonal and specialization” method, following [CTP14], [Voi15b], and [Pir16]. A projective variety X over a ﬁeld k is universally CH0-trivial if for all ﬁe”
On the rationality problem for quadric bundles (2017, methodology): “nd [HPT3].ON THE RATIONALITY PROBLEM FOR QUADRIC BUNDLES 15 4. The specialization method via weak decompositions of the diagonal Recall from Section 1.2 that we aim to generalize the method of Voisin [Voi2] and Colliot-Th´el`ene–Pirutka [CTP1] to degenerations where the special ﬁbre is allowed to ha”
On the universal $CH_0$ group of cubic hypersurfaces (2014, background): “We prove similar results for very general nodal quartic double solids with k ≤ 7 nodes and in the case of very general double solids with exactly 7 nodes, we prove in [28] that the non-existence of a cohomological decomposition of the diagonal is equivalent to the non-existence of a universal codime”
Hypersurfaces that are not stably rational (2015, background): “Voisin made a breakthrough in 2013 by showing that a very general quartic double solid (a double cover of P ramified over a quartic surface) is not stably rational [23].”