Abstract. We prove that Landau-Ginzburg models for all 17 smooth Fano threefolds with Picard rank 1 can be represented as Laurent polynomials in 3 variables exhibiting them case by case. We check that these Landau-Ginzburg models can be compactified to open Calabi-Yau varieties. In the spirit of L. Katzarkov's program we prove that numbers of irreducible components of the central fibers of compactifications of these pencils are dimensions of intermediate Jacobians of Fano varieties plus 1. In particular these numbers do not depend on compactifications. We state most of known methods of finding Landau-Ginzburg models in terms of Laurent polynomials. We discuss Laurent polynomial representation of Landau-Ginzburg models of Fano varieties and state some problems related to it.

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We show that every Picard rank one smooth Fano threefold has a weak Landau-Ginzburg model coming from a toric degeneration. The fibers of these Landau-Ginzburg models can be compactified to K3 surfaces with Picard lattice of rank 19. We also show that any smooth Fano variety of arbitrary dimension which is a complete intersection of Cartier divisors in weighted projective space has a very weak Landau-Ginzburg model coming from a toric degeneration.

Abstract. We prove that smooth Fano threefolds have toric Landau-Ginzburg models. More precise, we prove that their Landau-Ginzburg models, presented as Laurent polynomials, admit compactifications to families of K3 surfaces, and we describe their fibers over infinity. We also give an explicit construction of Landau-Ginzburg models for del Pezzo surfaces and any divisors on them.

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...