TY - JOUR

T1 - Double solids, categories and non-rationality

AU - Iliev, Atanas

AU - Katzarkov, Ludmil

AU - Przyjalkowski, Victor

N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 2014/2

Y1 - 2014/2

N2 - 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.

AB - 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.

KW - Fano varieties

KW - Landau-Ginzburg model

KW - rationality questions

UR - http://www.scopus.com/inward/record.url?scp=84897028388&partnerID=8YFLogxK

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U2 - 10.1017/S0013091513000898

DO - 10.1017/S0013091513000898

M3 - Article

AN - SCOPUS:84897028388

VL - 57

SP - 145

EP - 173

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

SN - 0013-0915

IS - 1

ER -