Double Solids, Categories and Non-Rationality

Abstract

Abstract. This paper suggests a new approach to questions of rationality of threefolds based on category theory. Following [BFK10] and [BFK11] we enhance constructions from [Kuz09] by introducing NoetherLefschetz spectra -an interplay between Orlov spectra [Ol94] and Hochschild homology. The main goal of this paper is to suggest a series of interesting examples where above techniques might apply. We start by constructing a sextic double solid X with 35 nodes and torsion in H 3 (X, Z). This is a novelty -after the classical example of Artin and Mumford (1972), this is the second example of a Fano threefold with a torsion in the 3-rd integer homology group. In particular X is non-rational. We consider other examples as well -V 10 with 10 singular points and double covering of quadric ramified in octic with 20 nodal singular points.After analyzing the geometry of their Landau Ginzburg models we suggest a general non-rationality picture based on Homological Mirror Symmetry and category theory. IntroductionThis paper suggests a new approach to questions of rationality of threefolds based on category theory. It was inspired by recent work of V. Shokurov and by A. Kuznetsov's idea about the Griffiths component (see [Kuz08]). This work is a natural continuation of ideas developed in [Ka09], [GKKN11] and of ideas of Kawamata and his school.We first extend classical example of Artin and Mumford to construct a sextic double solid X with 35 nodes and torsion in H 3 (X, Z). The construction is based on an approach by M. Gross and suggests close relation between Artin and Mumford example and the sextic double solid X with 35 nodes. This example, a novelty on its own, opens a possibility of series of interesting examples -V 10 with 10 singular points and double covering of quadric ramified in octic with 20 nodal singular points.In this paper we start investigating these examples from the point of view of Homological Mirror Symmetry (HMS). We consider the mirrors of the sextic double solid X with 35 nodes, of the Fano variety V 10 with 10 singular points in general position and of the double covering of quadric ramified in octic with 20 nodal singular points. We note that the monodromy around the singular fiber over zero of the Landau-Ginzburg models is strictly unipotent in all these examples, which suggests that the categorical behavior should be very similar to the one of the Artin-Mumford example. We conjecture that the reason for categorical similarity in all these examples is that they contained the category of an Enriques surface as a semiorthogonal summand in their derived categories. This is done in Section 5, where we introduce Landau-Ginzburg models and compare their singularities.In Section 6 we introduce several new rationality invariants coming out of the notions of spectra and enhanced Noether-Lefschetz spectra of categories. We give a conjectural categorical explanation of the examples from Sections 2, 3, 4, 5. The novelty (conjecturally) is that non-rationality of these examples cannot be picked by Orlov spectr...

Birational Geometry via Moduli Spaces

“…Monodromy of Landau-Ginzburg models, gap in spectra is at most 2. This conjecture was partially verified in [KP09], [IKP11], [IKS]. We record our findings in Table 6.…”

“…We also introduce "higher" nonrationality categorical invariants -gaps of spectra, which conjecturally are not present in the Sarkisov examples. These ideas are natural continuation of [IKP11]. 5.…”

Categorical representability and intermediate Jacobians of Fano threefolds

“…The relation between torsion in H 3 (X, Z) and categorical representability needs a further investigation, for example in the case recently described in [IKP11]. (Y, B).…”

“…Finally, conjectures based on homological mirror symmetry are proposed in [Kat09,Kat10], but we cannot state a precise relation with our construction. A careful study of the example constructed in [IKP11] would be a good starting point.…”

The rationality problem for conic bundles

“…This example was generalized in [IKP14] and [PS16d] (see also [Bea16]) to some other types of singular Fano threefolds.…”