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Enriques–Kodaira classification

In mathematics, the Enriques–Kodaira classification groups compact complex surfaces into ten classes, each parametrized by a moduli space. For most of the classes the moduli spaces are well understood, but for the class of surfaces of general type the moduli spaces seem too complicated to describe explicitly, though some components are known.

Characters, Invariants of surfaces & Classification in positive characteristic

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Invariants of surfaces

Coherent sheaf cohomology Plurigenera Canonical line bundle Birational Seiberg–Witten theory

33 related topics

Classification in positive characteristic

DavidMumford 1977 EnricoBombieri Picard scheme Quasi-elliptic surfaces

10 related topics

Surfaces of Kodaira dimension −∞

Rational surface Complex projective plane Hirzebruch surfaces Quadrics Cubic surfaces

12 related topics

Surfaces of Kodaira dimension 0

Intersection number with itself Unimodular lattice Kummer surfaces Complex tori Abelian surfaces

10 related topics

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Overview

Statement of the classification

Global spherical shell conjecture

Invariants of surfaces

Minimal models and blowing up

Surfaces of Kodaira dimension −∞

Rational surface
Complex projective plane
Hirzebruch surfaces Hirzebruch surface
Quadrics Quadric
Cubic surfaces Cubic surface
Del Pezzo surfaces Del Pezzo surface
Veronese surface
Hopf surfaces Hopf surface
Inoue surfaces Inoue surface
Inoue-Hirzebruch surfaces Inoue-Hirzebruch surface
Enoki surfaces Enoki surface
Kato surfaces Kato surface

Surfaces of Kodaira dimension 0

Intersection number with itself Self-intersection number
Unimodular lattice
Kummer surfaces Kummer surface
Complex tori
Abelian surfaces Abelian surface
Abelian varieties Abelian variety
Isogeny
Complex number
Enriques surfaces Enriques surface
Hyperelliptic surfaces Hyperelliptic surface

Surfaces of Kodaira dimension 1

Complete list of the possible singular fibers Elliptic surface
Discrete valuation rings Discrete valuation ring
P-adic integers P-adic integer
Dedekind domains Dedekind domain
Kodaira surfaces Kodaira surface
K3 surfaces K3 surface

Surfaces of Kodaira dimension 2 (surfaces of general type)

Coarse moduli scheme
Bogomolov–Miyaoka–Yau inequality
Hilbert modular surfaces Hilbert modular surface
Fake projective planes Fake projective plane
Barlow surfaces Barlow surface

Classification in positive characteristic

DavidMumford David Mumford
1977 Enriques–Kodaira classification
EnricoBombieri Enrico Bombieri
Picard scheme
Quasi-elliptic surfaces Quasi-elliptic surface
Group schemes Group scheme
Étale Étale group scheme
Oscar Zariski
Inseparable extensions Inseparable extension
Zariski surfaces Zariski surface

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Enriques–Kodaira classification

Nodes98

Edges97

Triples6

Avg. degree1.98

Density0.020408

Components1

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Enriques–Kodaira classification

Top relations

related to Statement of the classification · 5

Enriques–Kodaira classification → For, Kodaira, The, The Enriques, VII

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Important terminology

surfaces kodaira surface complex algebraic dimension classification displaystyle enriques invariants minimal characteristic group plurigenera elliptic hodge genus compact examples numbers

Entity relationships Subject–Predicate–Object triples

Subject	Predicate	Object	Confidence	Src
the Picard group Pic	instance of	These include algebraic invariants	0.80	text
Enriques–Kodaira classification	related to Statement of the classification	The Enriques	0.60	section
Enriques–Kodaira classification	related to Statement of the classification	Kodaira	0.60	section
Enriques–Kodaira classification	related to Statement of the classification	The	0.60	section
Enriques–Kodaira classification	related to Statement of the classification	For	0.60	section
Enriques–Kodaira classification	related to Statement of the classification	VII	0.60	section

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