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
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben

2018 | Buch

Preliminaries Kapitel lesen Erstes Kapitel lesen

Algebraic Curves

Towards Moduli Spaces

verfasst von: Prof. Dr. Maxim E. Kazaryan, Prof. Dr. Sergei K. Lando, Prof. Dr. Victor V.  Prasolov

Verlag: Springer International Publishing

Buchreihe : Moscow Lectures

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

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Über dieses Buch

This book offers a concise yet thorough introduction to the notion of moduli spaces of complex algebraic curves. Over the last few decades, this notion has become central not only in algebraic geometry, but in mathematical physics, including string theory, as well.

The book begins by studying individual smooth algebraic curves, including the most beautiful ones, before addressing families of curves. Studying families of algebraic curves often proves to be more efficient than studying individual curves: these families and their total spaces can still be smooth, even if there are singular curves among their members. A major discovery of the 20th century, attributed to P. Deligne and D. Mumford, was that curves with only mild singularities form smooth compact moduli spaces. An unexpected byproduct of this discovery was the realization that the analysis of more complex curve singularities is not a necessary step in understanding the geometry of the moduli spaces.

The book does not use the sophisticated machinery of modern algebraic geometry, and most classical objects related to curves – such as Jacobian, space of holomorphic differentials, the Riemann-Roch theorem, and Weierstrass points – are treated at a basic level that does not require a profound command of algebraic geometry, but which is sufficient for extending them to vector bundles and other geometric objects associated to moduli spaces. Nevertheless, it offers clear information on the construction of the moduli spaces, and provides readers with tools for practical operations with this notion.

Based on several lecture courses given by the authors at the Independent University of Moscow and Higher School of Economics, the book also includes a wealth of problems, making it suitable not only for individual research, but also as a textbook for undergraduate and graduate coursework

Inhaltsverzeichnis

Frontmatter
Chapter 1. Preliminaries
Abstract
The main focus of this course will be on complex algebraic curves, or, which is the same, Riemann surfaces. From the topological point of view, a Riemann surface is a two-dimensional oriented surface; its topological properties are uniquely determined by a nonnegative integer, the genus. At the same time, individual characteristics of algebraic curves are complicated, and two different curves, even of the same genus, usually bear little resemblance to each other. However, if we look at curves not one by one, but in families, it turns out that such families have a relatively simple structure and many remarkable properties, which find various applications in mathematics and theoretical physics. It is the transition from individual curves to families of curves that is the topic of the course.
Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
Chapter 2. Algebraic Curves
Abstract
Algebraic curves are curves given by polynomial equations in projective spaces. On the other hand, algebraic curves are one-dimensional complex manifolds, and to define them, there is no need to embed them anywhere. We will consider various ways to define curves and discuss how one can decide whether they result in the same curve.
Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
Chapter 3. Complex Structure and the Topology of Curves
Abstract
Every complex algebraic curve is a two-dimensional oriented surface. As we already know, the topology of such surfaces is very simple: for a compact surface, the topology is uniquely determined by its genus (or, equivalently, its Euler characteristic). However, along with a topological structure, a curve has a complex structure. It singles out analytic functions among all the functions on the curve.
Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
Chapter 4. Curves in Projective Spaces
Abstract
As we will see below, the projective plane (like any other two-dimensional surface) is too narrow to accommodate every smooth curve. In the three-dimensional space there is much more freedom. However, to define curves in \({\mathbb {C}\mathbb {P}}^3\) and higher dimensional projective spaces is more difficult than in the plane. In this chapter, we discuss methods of defining such curves.
Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
Chapter 5. Plücker Formulas
Abstract
The plane curves break into pairs: with each plane curve one can associate a dual curve, the dual of the dual coinciding with the original curve. Usually, the dual of a smooth curve turns out to be singular, hence, studying duality, we cannot content ourselves with considering only smooth curves. Moreover, it does not suffice to consider only curves with singularities of simplest form, points of transversal self-intersection. However, the pairs of dual curves having only points of transversal self-intersection and cusps form an open subset in the space of pairs of dual curves of given degrees, which makes it natural to study such pairs. Plücker formulas are relations on the number of singularities of various types for a pair of dual curves of given degrees.
Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
Chapter 6. Mappings of Curves
Abstract
A meromorphic function on an algebraic curve is a mapping from this curve into the projective line. However, it is natural to consider also mappings into other complex curves, first of all, one-to-one mappings from a complex curve to itself, i.e., automorphisms of a curve. All automorphisms of a given curve form a group. For a curve of genus 0 (projective line), this group is three-dimensional. For any curve of genus 1 (elliptic curve), it is one-dimensional. For curves of higher genus it is finite, and for curves of genus g > 2 it usually consists only of the identity mapping. Curves with a large symmetry group are of special interest: like any symmetric object, they can be very beautiful.
Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
Chapter 7. Differential 1-Forms on Curves
Abstract
It is convenient to express the basic properties of curves and, more generally, arbitrary complex manifolds, in terms of various related objects. We mean primarily spaces of meromorphic functions, vector fields, and differential forms. These spaces are endowed with natural algebraic structures, which allows one to express properties of curves in algebraic terms.
Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
Chapter 8. Line Bundles, Linear Systems, and Divisors
Abstract
We have already come across line bundles over curves: trivial, tangent, and cotangent bundles and their tensor powers. Thus, a natural question arises: given a curve, are there other line bundles over it? For instance, in the case of elliptic curves, all line bundles mentioned above are trivial, so this question means, in particular, whether there are nontrivial line bundles over a given elliptic curve.
Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
Chapter 9. Riemann–Roch Formula and Its Applications
Abstract
The Riemann–Roch theorem establishes a relationship between two numbers: the dimension l(D) of the vector space L(D) of meromorphic functions with divisor ≥−D and the dimension i(D) of the space Ω 1(D) of meromorphic 1-forms with divisor ≥ D.
Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
Chapter 10. Proof of the Riemann–Roch Formula
Abstract
In the first section of this chapter, we give a proof of the Riemann–Roch formula l(D) − l(K − D) = d − g + 1. In the second section, we present a geometric interpretation of the quantities occurring in the Riemann–Roch formula in terms of canonical curves.
Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
Chapter 11. Weierstrass Points
Abstract
On curves of genus g ≥ 2, points differ from each other. For example, every nontrivial automorphism of such a curve has finitely many fixed points, and the set of points that are fixed by some nontrivial automorphism is also finite. In this chapter, we discuss another method of singling out points with special properties. This method, due to Weierstrass, also singles out a finite set of special points on every curve. This set is closely related to the set of fixed points of nontrivial automorphisms, but does not in general coincide with it. In particular, Weierstrass points on a curve do exist even if it has no nontrivial automorphisms.
Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
Chapter 12. Abel’s Theorem
Abstract
The line bundles over a given complex curve are in a one-to-one correspondence with the linear equivalence classes of divisors on this curve. Such a class has an integer-valued characteristic, the degree. Since divisors consist of points of the curve, it is natural to expect that the set of classes of divisors of the same degree is endowed with additional structures. It must be a topological space and, moreover, a complex variety. Abel’s theorem identifies the space of classes of divisors of zero degree on a curve of genus g with a g-dimensional complex torus, the Jacobian of the curve.
Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
Chapter 13. Examples of Moduli Spaces
Abstract
In Sect. 6.​3, we have already discussed what does the moduli space of elliptic (i.e., genus 1) curves look like. This is a rather typical example, which allows one to observe many features common for all moduli spaces. In this chapter, we will study it in more detail and consider examples of other moduli spaces. From the very beginning, we deal with moduli spaces of curves with marked points: properties of such spaces can be described “in layman’s terms” already for curves of small genus, and these properties are highly nontrivial. As the genus grows, the geometry of the moduli space becomes complicated, and a complete description of this space is usually beyond reach. Nevertheless, one can compute many important geometric characteristics of moduli spaces of curves of high genus.
Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
Chapter 14. Approaches to Constructing Moduli Spaces
Abstract
Constructing moduli spaces is a technically complicated task, involving the analysis of many subtleties. In this chapter, we will discuss, without going into details, one of the possible methods of carrying out such a construction. We will describe the general sequence of steps; numerous results justifying it are either stated without proofs or left as exercises.
Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
Chapter 15. Moduli Spaces of Rational Curves with Marked Points
Abstract
In the previous chapter, we have described a procedure for constructing moduli spaces of curves with marked points. These spaces are usually not compact. The key point in the understanding of the geometry of noncompact spaces is constructing their compactifications. Every noncompact space has many different compactifications, but only few of them are convenient to work with. In contrast to higher genera, the moduli space \(\overline {{\mathcal {M}}_{0;n}}\) is a smooth manifold rather than orbifold. This property simplifies a bit investigation of moduli spaces in genus 0 case.
Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
Chapter 16. Stable Curves
Abstract
In the previous chapter, we introduced the notion of a stable rational curve with marked points. The (modular) stability of a curve means that it has a finite group of automorphisms.
Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
Chapter 17. A Backward Look from the Viewpoint of Characteristic Classes
Abstract
The introduction of characteristic classes allows one to see many calculations carried out above in a new light and simplify them. Characteristic classes are a universal tool for computing topological characteristics of algebraic varieties, both smooth and singular. We begin with discussing definitions and general properties of Chern classes of vector bundles, and then show how one can use them to obtain some results we already know and their generalizations. In the next chapter, we will speak about characteristic classes that arise naturally in the study of the topology of moduli spaces of curves.
Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
Chapter 18. Moduli Spaces of Stable Maps
Abstract
In this chapter, we show how moduli spaces of maps can be applied to compute topological characteristics of various varieties. The notion of a stable map was introduced by Kontsevich. He applied it to solving the classical problem of enumerating rational curves of a given degree in the plane passing through a given collection of points. The methods suggested by Kontsevich turned out to be applicable to a wide circle of problems of enumerative geometry, being now the main tool for computing Gromov–Witten invariants.
Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
Chapter 19. Exam Problems
Abstract
Below we give a list of problems used in final exams in the 1-year course repeatedly taught by M. E. Kazaryan and S. K. Lando at the Department of Mathematics of the Higher School of Economics in 2010–2014. Most of these problems were given as exercises in the main text, and we have collected them here for the reader’s convenience. Along with problems, we also give a list of exam questions.
Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
Backmatter
Metadaten
Titel
Algebraic Curves
verfasst von
Prof. Dr. Maxim E. Kazaryan
Prof. Dr. Sergei K. Lando
Prof. Dr. Victor V. Prasolov
Copyright-Jahr
2018
Electronic ISBN
978-3-030-02943-2
Print ISBN
978-3-030-02942-5
DOI
https://doi.org/10.1007/978-3-030-02943-2