Birational Geometry via Moduli Spaces

Cheltsov, Ivan; Katzarkov, Ludmil; Przyjalkowski, Victor

doi:10.1007/978-1-4614-6482-2_5

Cited by 12 publications

(12 citation statements)

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“…The final picture is obtained by gluing the configurations of surfaces shown in Figures 8-10 along Figure 11. A more detailed description of the Landau-Ginzburg model for a sextic double solid can be found in [8]. Direct calculations (see [25,27]) yield the following.…”

Section: The Landau-ginzburg Model Of a Sextic Double Solidmentioning

confidence: 99%

Double Solids, Categories and Non-Rationality

Iliev

Katzarkov²,

Przyjalkowski

2013

Proceedings of the Edinburgh Mathematical Society

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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...

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Section: The Landau-ginzburg Model Of a Sextic Double Solidmentioning

confidence: 99%

Double Solids, Categories and Non-Rationality

Iliev

Katzarkov²,

Przyjalkowski

2013

Proceedings of the Edinburgh Mathematical Society

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“…Unlike the two-dimensional case, there is no structure in the list of Fano threefolds (see [46]) relating one with each other systematically. An approach to get such structure is given in [15]. The idea behind the approach is the following.…”

Section: Introductionmentioning

confidence: 99%

“…Thus one can construct, in terms of spanning polytopes the needed projections. An example of nice subtree in the projections tree relating Picard rank one Fano varieties (Figure 1) is found in [15]. Moreover, one can implement mutations in the picture (see Section 4), that is deformations from one toric degeneration to another.…”

Section: Introductionmentioning

confidence: 99%

Projecting Fanos in the mirror

Kasprzyk,

Katzarkov,

Przyjalkowski

et al. 2019

Preprint

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In the paper "Birational geometry via moduli spaces" by I. Cheltsov, L. Katzarkov, and V. Przyjalkowski a new structure connecting toric degenerations of smooth Fano threefolds by projections was introduced; using Mirror Symmetry these connections were transferred to the side of Landau-Ginzburg models. In the paper mentioned above a nice way to connect of Picard rank one Fano threefolds was found. We apply this approach to all smooth Fano threefolds, connecting their degenerations by toric basic links. In particular, we find a lot of Gorenstein toric degenerations of smooth Fano threefolds we need. We implement mutations in the picture as well. It turns out that appropriate chosen toric degenerations of the Fanos are given by toric basic links from a few roots. We interpret the relations we found in terms of Mirror Symmetry.

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“…Настоящая работа является первым шагом к изучению слабых моделей Ландау-Гинзбурга трехмерных многообразий Фано. В работах [6], [8], [10] более детально изучены конкретные свойства некоторых слабых моделей Ландау-Гинзбурга и их связь с гомологической зеркальной симметрией.…”

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“…Доказать, что модель Ландау-Гинзбурга для полного пересечения в G(m, r) является слабой в общем случае (ср. проблемы10,14).…”

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Слабые Модели Ландау - Гинзбурга Гладких Трехмерных Многообразий Фано

Пржиялковский¹,

Przyjalkowski²

2013

Известия Российской академии наук. Серия математическая

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Рассмотрены модели Ландау-Гинзбурга для гладких трехмерных многообразий Фано основной серии и показано, что они представляются многочленами Лорана. Проверяется, что данные модели Ландау-Гинзбурга могут быть компактифицированы до открытых многообразий Калаби-Яу. В духе программы Л. Кацаркова показано, что числа неприводимых компонент центральных слоев компактификаций найденных пучков на единицу больше размерностей промежуточных якобианов соответствующих многообразий Фано. (В частности, эти числа не зависят от способа компактификации.) Сформулированы основные известные методы нахождения моделей Ландау-Гинзбурга как многочленов Лорана. Обсуждаются представления моделей Ландау-Гинзбурга многообразий Фано в виде многочленов Лорана и формулируются проблемы, связанные с таким представлением. Библиография: 44 наименования. Ключевые слова: слабые модели Ландау-Гинзбурга, многообразия Фано, торические вырождения, промежуточный якобиан.

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Birational Geometry via Moduli Spaces

Cited by 12 publications

References 84 publications

Double Solids, Categories and Non-Rationality

Double Solids, Categories and Non-Rationality

Projecting Fanos in the mirror

Слабые Модели Ландау - Гинзбурга Гладких Трехмерных Многообразий Фано

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