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Double Solids, Categories and Non-Rationality

Published online by Cambridge University Press:  23 December 2013

Atanas Iliev ,
Ludmil Katzarkov  and
Victor Przyjalkowski
Atanas Iliev
Affiliation:
Department of Mathematical Sciences, College of Natural Science, Seoul National University, San 56-1, Sillim-dong, Gwanak-gu, Seoul 151-747, Republic of Korea (ailiev2001@yahoo.com)
Ludmil Katzarkov
Affiliation:
Department of Mathematics, University of California, Irvine, CA 92697-3875, USA (lkatzark@math.uci.edu)
Victor Przyjalkowski
Affiliation:
Mathematical Institute, Russian Academy of Sciences, 32A Leninsky Avenue, Moscow, Russia (victorprz@mi.ras.ru; victorprz@gmail.com)
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Abstract

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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 H3(X, ℤ). 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: V10 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.

Keywords

Fano varietiesrationality questionsLandau–Ginzburg modelPrimary 14E0814F05Secondary 14J4514J33
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 1 , February 2014 , pp. 145 - 173
Copyright
Copyright © Edinburgh Mathematical Society 2014 

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