Double solids, categories and non-rationality

Atanas Iliev, Ludmil Katzarkov, Victor Przyjalkowski

Research output: Contribution to journalArticlepeer-review

15 Scopus citations

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 languageEnglish (US)
Pages (from-to)145-173
Number of pages29
JournalProceedings of the Edinburgh Mathematical Society
Volume57
Issue number1
DOIs
StatePublished - Feb 2014
Externally publishedYes

Keywords

  • Fano varieties
  • Landau-Ginzburg model
  • rationality questions

ASJC Scopus subject areas

  • Mathematics(all)

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