### Abstract

This paper suggests a new approach to questions of rationality of 3-folds based on category theory. Following work by Ballard et al., we enhance constructions of Kuznetsov by introducing Noether-Lefschetz spectra: an interplay between Orlov spectra and Hochschild homology. The main goal of this paper is to suggest a series of interesting examples where the above techniques might apply. We start by constructing a sextic double solid X with 35 nodes and torsion in H ^{3}(X, â). This is a novelty: after the classical example of Artin and Mumford, this is the second example of a Fano 3-fold with a torsion in the third integer homology group. In particular, X is non-rational. We consider other examples as well: V 10 with 10 singular points, and the double covering of a quadric ramified in an octic with 20 nodal singular points. After analysing the geometry of their Landau-Ginzburg models, we suggest a general non-rationality picture based on homological mirror symmetry and category theory.

Original language | English (US) |
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Pages (from-to) | 145-173 |

Number of pages | 29 |

Journal | Proceedings of the Edinburgh Mathematical Society |

Volume | 57 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2014 |

Externally published | Yes |

### Keywords

- Fano varieties
- Landau-Ginzburg model
- rationality questions

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Proceedings of the Edinburgh Mathematical Society*,

*57*(1), 145-173. https://doi.org/10.1017/S0013091513000898