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Double solids, categories and non-rationality. (English) Zbl 1303.14026
The authors present an example of a complex Fano threefold \(X\) whose non-rationality is detected by the torsion in \(H^3(X,\mathbb{Z})\).
The authors consider a sextic double solid \(X\) with 35 nodes, whose branch locus is a sextic surface in \(\mathbb{P}^3\) obtained as the degeneration divisor of a quadric fibration \(W \to \mathbb{P}^3\) constructed as a section of the Segre variety \(\mathbb{P}^3 \times \mathbb{P}^4 \subset \mathbb{P}^{19}\) by a hyperplane and a divisor of bidegree \((1,2)\). The quadratic form defined by \(W\) gives a nontrivial Brauer class in \(\mathrm{Br}(X)\) providing a torsion element in \(H^3(X,\mathbb{Z})\). Similarly, they describe a determinantal quartic double solid and show its non-rationality. Some mirror-symmetry theoretical considerations lead the authors to consider Fano threefolds of type \(V_{10}\) and their Landau–Ginzburg models in connection with quartic double solids, as well as Landau–Ginzburg models of sextic double solids.
In the last section, the authors define the enhanced Noether–Lefschetz spectrum of a dg-enhanced triangulated category and propose some conjectures relating such spectra to rationality questions.

MSC:
14E08 Rationality questions in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J45 Fano varieties
14J33 Mirror symmetry (algebro-geometric aspects)
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