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2020 | Buch

Birational Geometry and Moduli Spaces

herausgegeben von: Prof. Elisabetta Colombo, Prof. Barbara Fantechi, Prof. Paola Frediani, Prof. Donatella Iacono, Prof. Rita Pardini

Verlag: Springer International Publishing

Buchreihe : Springer INdAM Series

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

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Über dieses Buch

This volume collects contributions from speakers at the INdAM Workshop “Birational Geometry and Moduli Spaces”, which was held in Rome on 11–15 June 2018. The workshop was devoted to the interplay between birational geometry and moduli spaces and the contributions of the volume reflect the same idea, focusing on both these areas and their interaction. In particular, the book includes both surveys and original papers on irreducible holomorphic symplectic manifolds, Severi varieties, degenerations of Calabi-Yau varieties, uniruled threefolds, toric Fano threefolds, mirror symmetry, canonical bundle formula, the Lefschetz principle, birational transformations, and deformations of diagrams of algebras. The intention is to disseminate the knowledge of advanced results and key techniques used to solve open problems. The book is intended for all advanced graduate students and researchers interested in the new research frontiers of birational geometry and moduli spaces.

Inhaltsverzeichnis

Frontmatter

Negative Rational Curves and Their Deformations on Hyperkähler Manifolds

Abstract

We survey some results about rational curves on hyperkähler manifolds, explaining how to prove a certain deformation-invariance statement for loci covered by rational curves with negative Beauville–Bogomolov square.

Ekaterina Amerik

Moduli Spaces of Cubic Threefolds and of Irreducible Holomorphic Symplectic Manifolds

Abstract

In this survey, based on joint work of the author and S. Boissière and A. Sarti, we will describe an isomorphism between the moduli space of smooth cubic threefolds, as described by Allcock, Carlson and Toledo, and the moduli space of fourfolds of K3^[2]-type with a special non-symplectic automorphism of order three; then, I will show some consequences of this isomorphism concerning degenerations of non-symplectic automorphisms. Finally we will explore possible generalizations of the problem to higher dimensions and other moduli spaces of cubic threefolds.

Chiara Camere

A Note on Severi Varieties of Nodal Curves on Enriques Surfaces

Abstract

Let |L| be a linear system on a smooth complex Enriques surface S whose general member is a smooth and irreducible curve of genus p, with L ² > 0, and let V _|L|,δ(S) be the Severi variety of irreducible δ-nodal curves in |L|. We denote by π : X → S the universal covering of S. In this note we compute the dimensions of the irreducible components V of V _|L|,δ(S). In particular we prove that, if C is the curve corresponding to a general element [C] of V , then the codimension of V in |L| is δ if π ⁻¹(C) is irreducible in X and it is δ − 1 if π ⁻¹(C) consists of two irreducible components.

Ciro Ciliberto, Thomas Dedieu, Concettina Galati, Andreas Leopold Knutsen

A Travel Guide to the Canonical Bundle Formula

Abstract

We survey known results on the canonical bundle formula and its applications in algebraic geometry.

Enrica Floris, Vladimir Lazić

Some Examples of Calabi–Yau Pairs with Maximal Intersection and No Toric Model

Abstract

It is known that a maximal intersection log canonical Calabi–Yau surface pair is crepant birational to a toric pair. This does not hold in higher dimension: this article presents some examples of maximal intersection Calabi–Yau pairs that admit no toric model.

Anne-Sophie Kaloghiros

On Deformations of Diagrams of Commutative Algebras

Abstract

In this paper we study classical deformations of diagrams of commutative algebras over a field of characteristic 0. In particular we determine several homotopy classes of DG-Lie algebras, each one of them controlling this above deformation problem: the first homotopy type is described in terms of the projective model structure on the category of diagrams of differential graded algebras, the others in terms of the Reedy model structure on truncated Bousfield-Kan approximations.

The first half of the paper contains an elementary introduction to the projective model structure on the category of commutative differential graded algebras, while the second half is devoted to the main results.

Emma Lepri, Marco Manetti

The Lefschetz Principle in Birational Geometry: Birational Twin Varieties

Abstract

Inspired by the Weak Lefschetz Principle, we study when a smooth projective variety fully determines the birational geometry of some of its subvarieties. In particular, we consider the natural embedding of the space of complete quadrics into the space of complete collineations and we observe that their birational geometry, from the point of view of Mori theory, fully determines each other. When two varieties are related in this way, we call them birational twins. We explore this notion and its various flavors for other embeddings between Mori dream spaces.

César Lozano Huerta, Alex Massarenti

What is the Monodromy Property for Degenerations of Calabi-Yau Varieties?

Abstract

In this survey, we discuss the state of art about the monodromy property for Calabi-Yau varieties. We explain what is the monodromy property for Calabi-Yau varieties and then discuss some examples of Calabi-Yau varieties that satisfy this property. After this recap, we discuss a possible approach to future research in this area.

Luigi Lunardon

Examples of Irreducible Symplectic Varieties

Abstract

Irreducible symplectic manifolds are one of the three building blocks of compact Kähler manifolds with numerically trivial canonical bundle by the Beauville-Bogomolov decomposition theorem. There are several singular analogues of irreducible symplectic manifolds, in particular in the context of compact Kähler orbifolds, and in the context of normal projective varieties with canonical singularities. In this paper we will collect their definitions, analyze their mutual relations and provide a list of known examples.

Arvid Perego

An Example of Mirror Symmetry for Fano Threefolds

Abstract

In this note we illustrate the Fanosearch programme of Coates, Corti, Galkin, Golyshev, and Kasprzyk in the example of the anticanonical cone over the smooth del Pezzo surface of degree 6.

Andrea Petracci

Chern Numbers of Uniruled Threefolds

Abstract

In this paper we show that the Chern numbers of a smooth Mori fibre space in dimension three are bounded in terms of the underlying topological manifold. We also generalise a theorem of Cascini and the second named author on the boundedness of Chern numbers of certain threefolds to the case of negative Kodaira dimension.

Stefan Schreieder, Luca Tasin

Titel: Birational Geometry and Moduli Spaces
herausgegeben von: Prof. Elisabetta Colombo
Prof. Barbara Fantechi
Prof. Paola Frediani
Prof. Donatella Iacono
Prof. Rita Pardini
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-37114-2
Print ISBN: 978-3-030-37113-5
DOI: https://doi.org/10.1007/978-3-030-37114-2