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Double Solids, Categories and Non-Rationality
Iliev, Atanas,Katzarkov, Ludmil,Przyjalkowski, Victor Cambridge University Press 2014 Proceedings of the Edinburgh Mathematical Society Vol.57 No.1
<B>Abstract</B><P>This paper suggests a new approach to questions of rationality of 3-folds based on category theory. Following work by Ballard <I>et al.</I>, 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 <I>X</I> with 35 nodes and torsion in <I>H</I><SUP>3</SUP>(<I>X, ℤ</I>). 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, <I>X</I> is non-rational. We consider other examples as well: <I>V</I>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.</P>