Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Bücher
Zeitschriften
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
MTZ-Fachkongress

Antriebe und Energiesysteme von morgen


Austausch von Automobil- und Energiewirtschaft

Am 14./15. Mai geht es in Chemnitz um die Mobilitätswende durch Elektro- und Wasserstofffahrzeuge.
Erweiterte Suche
Anmelden
nach oben

2021 | Buch

On Geometry of Fano Threefold Hypersurfaces Kapitel lesen Erstes Kapitel lesen

Rationality of Varieties

herausgegeben von: Gavril Farkas, Gerard van der Geer, Mingmin Shen, Lenny Taelman

Verlag: Springer International Publishing

Buchreihe : Progress in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Über dieses Buch

This book provides an overview of the latest progress on rationality questions in algebraic geometry. It discusses new developments such as universal triviality of the Chow group of zero cycles, various aspects of stable birationality, cubic and Fano fourfolds, rationality of moduli spaces and birational invariants of group actions on varieties, contributed by the foremost experts in their fields.

The question of whether an algebraic variety can be parametrized by rational functions of as many variables as its dimension has a long history and played an important role in the history of algebraic geometry. Recent developments in algebraic geometry have made this question again a focal point of research and formed the impetus to organize a conference in the series of conferences on the island of Schiermonnikoog. The book follows in the tradition of earlier volumes, which originated from conferences on the islands Texel and Schiermonnikoog.

Inhaltsverzeichnis

Frontmatter
On Geometry of Fano Threefold Hypersurfaces
Abstract
We prove that a quasi-smooth Fano threefold hypersurface is birationally rigid if and only if it has Fano index one.
Hamid Abban, Ivan Cheltsov, Jihun Park
On the Image of the Second l-adic Bloch Map
Abstract
For a smooth projective geometrically uniruled threefold defined over a perfect field we show that there exists a canonical abelian variety over the field, namely the second algebraic representative, whose rational Tate modules model canonically the third l-adic cohomology groups of the variety for all primes l. In addition, there exists a rational correspondence inducing these identifications. In the case of a geometrically rationally chain connected variety, one obtains canonical identifications between the integral Tate modules of the second algebraic representative and the third l-adic cohomology groups of the variety, and if the variety is a geometrically stably rational threefold, these identifications are induced by an integral correspondence. Our overall strategy consists in studying – for arbitrary smooth projective varieties – the image of the second l-adic Bloch map restricted to the Tate module of algebraically trivial cycle classes in terms of the “correspondence (co)niveau filtration”. This complements results with rational coefficients due to Suwa. In the appendix, we review the construction of the Bloch map and its basic properties.
Jeffrey D. Achter, Sebastian Casalaina-Martin, Charles Vial
Rational Curves and MBM Classes on Hyperkähler Manifolds: A Survey
Abstract
This paper deals with rational curves and birational contractions on irreducible holomorphically symplectic manifold. We survey some recent results about minimal rational curves, their deformations, extremal rays associated with these curves, and the geometry of the Kähler cone.
Ekaterina Amerik, Misha Verbitsky
Unirationality of Certain Universal Families of Cubic Fourfolds
Abstract
The aim of this short note is to define the universal cubic fourfold over certain loci of their moduli space. Then, we propose two methods to prove that it is unirational over the Hassett divisors \( \mathcal{C}_{d} \), in the range 8 ≤ d ≤ 42. By applying inductively this argument, we are able to show that, in the same range of values, \( \mathcal{C}_{d,n} \) is unirational for all integer values of n. Finally, we observe that for explicit infinitely many values of d, the universal cubic fourfold over \( \mathcal{C}_{d} \) can not be unirational.
Hanine Awada, Michele Bolognesi
A Categorical Invariant for Geometrically Rational Surfaces with a Conic Bundle Structure
Abstract
We define a categorical birational invariant for minimal geometrically rational surfaces with a conic bundle structure over a perfect field via components of a natural semiorthogonal decomposition. Together with the similar known result on del Pezzo surfaces, this provides a categorical birational invariant for geometrically rational surfaces.
Marcello Bernardara, Sara Durighetto
Marked and Labelled Gushel–Mukai Fourfolds
Abstract
We prove that the moduli stacks of marked and labelled Hodgespecial Gushel–Mukai fourfolds are isomorphic. As an application, we construct rational maps from the stack of Hodge-special Gushel–Mukai fourfolds of discriminant d to the moduli space of (twisted) degree-d polarized K3 surfaces. We use these results to prove a counting formula for the number of fourdimensional fibers of Fourier–Mukai partners of very general Hodge-special Gushel–Mukai fourfolds with associated K3 surface, and a lower bound for this number in the case of a twisted associated K3 surface.
Emma Brakkee, Laura Pertusi
Supersingular Irreducible Symplectic Varieties
Abstract
In complex geometry, irreducible holomorphic symplectic varieties, also known as compact hyper-Kähler varieties, are natural higher-dimensional generalizations of K3 surfaces. We propose to study such varieties defined over fields of positive characteristic, especially the supersingular ones, generalizing the theory of supersingular K3 surfaces.
In this work, we are mainly interested in the following two types of symplectic varieties over an algebraically closed field of characteristic p > 0, under natural numerical conditions:
(1) smooth moduli spaces of semistable (twisted) sheaves on K3 surfaces,
(2) smooth Albanese fibers of moduli spaces of semistable sheaves on abelian surfaces.
Several natural definitions of the supersingularity for symplectic varieties are discussed, which are proved to be equivalent in both cases (1) and (2). Their equivalence is expected in general.
On the geometric side, we conjecture that unirationality characterizes supersingularity for symplectic varieties. Such an equivalence is established in case (1), assuming the same is true for K3 surfaces. In case (2), we show that rational chain connectedness is equivalent to supersingularity.
On the motivic side, we conjecture that algebraic cycles on supersingular symplectic varieties are much simpler than their complex counterparts: its rational Chow motive is of supersingular abelian type, the rational Chow ring is representable and satisfies the Bloch–Beilinson conjecture and Beauville’s splitting property. As evidence for this, we prove all these conjectures on algebraic cycles for supersingular varieties in both cases (1) and (2).
Lie Fu, Zhiyuan Li
Symbols and Equivariant Birational Geometry in Small Dimensions
Abstract
We discuss the equivariant Burnside group and related new invariants in equivariant birational geometry, with a special emphasis on applications in low dimensions.
Brendan Hassett, Andrew Kresch, Yuri Tschinkel
Rationality of Fano Threefolds of Degree 18 over Non-closed Fields
Abstract
We study unirationality and rationality of Fano threefolds of degree 18 over nonclosed fields.
Brendan Hassett, Yuri Tschinkel
Rationality of Mukai Varieties over Non-closed Fields
Abstract
We discuss birational properties of Mukai varieties, i.e., of higherdimensional analogues of prime Fano threefolds of genusg ∈ {7, 8, 9, 10} over an arbitrary field k of zero characteristic. In the case of dimension n ≥ 4 we prove that these varieties are k-rational if and only if they have a k-point except for the case of genus 9, where the same holds for n ≥ 5. Furthermore, we prove that Mukai varieties of genus g ∈ {7, 8, 9, 10} and dimension n ≥ 5 contain cylinders if they have a k-point. Finally, we prove that the embedding X ↪ Gr(3, 7) for prime Fano threefolds of genus 12 is defined canonically over any field of zero characteristic and use this to give a new proof of the criterion of k-rationality for these threefolds.
Alexander Kuznetsov, Yuri Prokhorov
A Refinement of the Motivic Volume, and Specialization of Birational Types
Abstract
We construct an upgrade of the motivic volume by keeping track of dimensions in the Grothendieck ring of varieties. This produces a uniform refinement of the motivic volume and its birational version introduced by Kontsevich and Tschinkel to prove the specialization of birational types. We also provide several explicit examples of obstructions to stable rationality arising from this technique.
Johannes Nicaise, John Christian Ottem
Explicit Rationality of Some Special Fano Fourfolds
Abstract
Recent results of Hassett, Kuznetsov and others pointed out countably many divisors \( \mathcal{C}_{d} \) in the open subset of \( \mathbb{P}^{55}= \mathbb{P}(\textit{H}^{o}(\mathcal{O}_{\mathbb{P}^{5}}(3))) \) parametrizing all cubic 4-folds and lead to the conjecture that the cubics corresponding to these divisors should be precisely the rational ones. Rationality has been proved by Fano for the first divisor \( \mathcal{C}_{14} \), in [RS19a] for the divisors \( \mathcal{C}_{26} \) and \( \mathcal{C}_{38} \), and in [RS19b] for \( \mathcal{C}_{42} \). In this note we describe explicit birational maps from a general cubic fourfold in \( \mathcal{C}_{14} \), in \( \mathcal{C}_{26} \) and in \( \mathcal{C}_{38} \) to P4, providing concrete geometric realizations of the more abstract constructions in [RS19a] and of the theoretical framework developed in [RS19b]. We also exhibit an explicit relationship between the divisor C14 and a certain divisor in the open subset of \( \mathbb{P}^{39}= \mathbb{P}(\textit{H}^{o}(\mathcal{O}_{Y}(2))) \) parametrizing smooth quadratic sections of a del Pezzo fivefold \( \textit{Y}=\mathbb{G}(1,4)\cap\mathbb{P}^{8}\subset\mathbb{P}^{8} \), the so-called Gushel–Mukai fourfolds.
Francesco Russo, Giovanni Staglianò
Unramified Cohomology, Algebraic Cycles and Rationality
Abstract
This is a survey on unramified cohomology with a view towards its applications to rationality problems.
Stefan Schreieder
Vanishing Cycles under Base Change and the Integral Hodge Conjecture
Abstract
In this paper we discuss an obstruction to the integral Hodge conjecture which arises from a certain behavior of vanishing cycles. This allows us to construct new counter-examples to the integral Hodge conjecture. One typical such counter-example is the product of a very general hypersurface of odd dimension and an Enriques surface. Our approach generalizes the degeneration argument of Benoist–Ottem [2].
Mingmin Shen
The Igusa Quartic and the Prym Map, with Some Rational Moduli
Abstract
This paper is devoted to the ubiquity of the Igusa quartic \( B \,\subset \,{\mathbb{P}^4} \) in connection to the Prym map \( \mathcal{p} \;:\, \mathcal{R}_{6} \rightarrow \mathcal{A}_{5} \). We introduce the moduli space \( \mathcal{X} \) of those quartic threefolds X cutting twice a quadratic section of B. A general X is 30-nodal and the intermediate Jacobian J(X) of its natural desingularization is a five-dimensional p.p. abelian variety. Let j :\( \mathcal{X}\rightarrow \mathcal{A}_{5} \) be the period map sending X to J(X), in the paper we study j and its relation to p. As is well known the degree of p is 27 and its monodromy group endows any smooth fibre F of p with the incidence configuration of 27 lines of a cubic surface. Then the same monodromy defines a map \( {j^\prime} \,:\, {\mathcal{D}_6}\, \rightarrow \, {\mathcal{A}_5}\) of degree 36, with fibre the configuration of 36 ’double-six’ sets of lines of a cubic surface. We prove that \( j = {j^\prime}\,o\, \phi, \, {\rm where} \, \phi \, : \, \mathcal{X}\,\rightarrow\,{\mathcal{D}_6}\) is birational. This provides a geometric description of \( {j^\prime}\). Finally we describe the relations between the different moduli spaces considered and prove that some, including \( \mathcal{X} \), are rational.
Alessandro Verra
Metadaten
Titel
Rationality of Varieties
herausgegeben von
Gavril Farkas
Gerard van der Geer
Mingmin Shen
Lenny Taelman
Copyright-Jahr
2021
Electronic ISBN
978-3-030-75421-1
Print ISBN
978-3-030-75420-4
DOI
https://doi.org/10.1007/978-3-030-75421-1

Premium Partner

    Bildnachweise