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401-4144-26L 4 Credits DR , MSC D-MATH

Complex Algebraic Surfaces

Lecturers & Examiners: Dr. Denis Nesterov
VVZ CR n/a

Last Updated: 2026-06-03 00:14:18

Abstract

Around the turn of the 20th century, Castelnuovo and Enriques undertook the classification of complex algebraic surfaces. Their work was later reformulated and completed in the modern language of algebraic geometry by Zariski, Kodaira, Shafarevich, and others. The course will present parts of this classification, with an emphasis on understanding the geometry and topology of algebraic surfaces.

Objective

The course aims to present the Enriques–Kodaira classification of complex algebraic surfaces, i.e., smooth complex projective varieties of dimension two. This classification organizes all such surfaces into eight geometrically distinct families and is based on a collection of discrete invariants that, in a certain sense, generalize the notion of genus for curves. To this end, we will first introduce the fundamental concepts required to study the geometry of algebraic surfaces, including: - Line bundles, divisors, and the Picard group - Serre's duality and Riemann-Roch theorem - Linear systems, rational maps, and blow-ups - Kodaira dimension We will then examine several important families of algebraic surfaces and study their properties using the aforementioned tools. Key examples appearing in the Enriques–Kodaira classification include: - Ruled surfaces - Rational surfaces - K3 and Abelian surfaces - Elliptic surfaces - Surfaces of general type ... Some surfaces will be covered in detail, illustrating algebro-geometric methods applicable in greater generality.

Content

We will closely follow the book "Complex Algebraic Surfaces" by A. Beauville. The book is based on a course taught by the author in Orsay and is organized as a series of lectures containing multiple exercises. Some preliminary material covering basic notions of algebraic geometry and topology will also be provided. The expectation is to cover the first five chapters of the book. The final lectures will be subject to choice depending on the audience’s preferences.

Resources

Literature

1. Beauville, A. (1996). Complex Algebraic Surfaces (2nd ed.). Cambridge: Cambridge University Press 2. Iskovskikh, V.A., Shafarevich, I.R. (1996). Algebraic Surfaces. In: Shafarevich, I.R. (eds) Algebraic Geometry II. Encyclopaedia of Mathematical Sciences, vol 35. Springer, Berlin, Heidelberg

Learning Materials (Links)

General Information

Language
English
Levels
DR , MSC

Examination

Type
session examination
Mode
oral 20 minutes
The exam is only offered in the summer 2026 and winter 2027 examination sessions.

Course Components

Type Title Time & Place Hours
lecture Complex Algebraic Surfaces
  • Thu 10:15-12:00 (ML F 34)
2 h weekly

Offered In