7.3 Algebraic surfaces and classification

Algebraic surfaces are two-dimensional complex projective varieties defined by polynomial equations. They're classified by their Kodaira dimension, which measures how their geometry grows. This classification helps us understand the structure and properties of different types of surfaces.

The Enriques-Kodaira classification divides surfaces into 10 classes based on their Kodaira dimension and other invariants. This system provides a complete picture of algebraic surfaces, from simple rational surfaces to complex surfaces of general type.

Algebraic surfaces and their invariants

Definition and basic properties of algebraic surfaces

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• An algebraic surface is a two-dimensional complex projective variety defined as the zero locus of a finite set of homogeneous polynomials in a projective space
• The dimension of an algebraic surface equals two, which is the maximum dimension of the tangent space at any smooth point
• The degree of an algebraic surface represents the number of intersection points with a general line in the projective space (e.g., a quadric surface has degree 2)
• Algebraic surfaces are compact complex manifolds of complex dimension 2

Important invariants of algebraic surfaces

• The canonical divisor of an algebraic surface represents the canonical sheaf, which is the sheaf of holomorphic 2-forms on the surface
  • It plays a crucial role in the classification of algebraic surfaces
• The arithmetic genus of an algebraic surface is a numerical invariant computed using the Hirzebruch-Riemann-Roch theorem
  • It depends on the canonical divisor and the Euler characteristic of the surface
• The geometric genus of an algebraic surface equals the dimension of the space of global holomorphic 2-forms on the surface
  • It is a birational invariant and is always less than or equal to the arithmetic genus
• The irregularity of an algebraic surface measures the difference between the arithmetic genus and the geometric genus
  • It vanishes for rational and ruled surfaces

Classifying algebraic surfaces

Classification by Kodaira dimension

• The Kodaira dimension is a numerical invariant that measures the growth rate of the plurigenera of an algebraic surface
  • Plurigenera are dimensions of spaces of global sections of tensor powers of the canonical sheaf
• Algebraic surfaces are classified into four categories based on their Kodaira dimension: κ = -∞, κ = 0, κ = 1, and κ = 2
  • Surfaces with κ = -∞ include ruled surfaces and rational surfaces
  • Surfaces with κ = 0 include abelian surfaces, K3 surfaces, Enriques surfaces, and bielliptic surfaces
  • Surfaces with κ = 1 are elliptic surfaces, which have a fibration over a curve with general fiber an elliptic curve
  • Surfaces with κ = 2 are of general type, and their canonical divisor is ample

Properties of surfaces in each Kodaira dimension

• Surfaces with κ = -∞:
  • Ruled surfaces are birational to the product of a curve and the projective line (e.g., Hirzebruch surfaces)
  • Rational surfaces are birational to the projective plane (e.g., cubic surfaces, del Pezzo surfaces)
• Surfaces with κ = 0:
  • Abelian surfaces are complex tori of dimension 2, with a group structure
  • K3 surfaces have trivial canonical bundle and are simply connected
  • Enriques surfaces are quotients of K3 surfaces by a fixed-point-free involution
  • Bielliptic surfaces are quotients of abelian surfaces by a finite group
• Surfaces with κ = 1:
  • Elliptic surfaces have a fibration over a curve with general fiber an elliptic curve
  • The base curve has genus at least 2, and the fibration has at least three singular fibers
• Surfaces with κ = 2:
  • Surfaces of general type have ample canonical divisor and are the most general class of algebraic surfaces
  • Examples include hypersurfaces of degree at least 5 in projective 3-space and complete intersections of sufficiently high multidegree

Geometry of ruled and rational surfaces

Ruled surfaces

• A ruled surface is an algebraic surface that is birational to the product of a curve and the projective line
• Every ruled surface can be obtained as the projectivization of a rank 2 vector bundle over a curve (e.g., the Hirzebruch surfaces $\mathbb{F}_n$ are projectivizations of $\mathcal{O} \oplus \mathcal{O}(n)$ over $\mathbb{P}^1$)
• The minimal model of a ruled surface is either:
  • The product of the projective line with itself (κ = -∞)
  • A minimal ruled surface over a curve of genus g ≥ 1 (κ = -∞ if g = 0, κ = 1 if g ≥ 1)
• Ruled surfaces have a rich geometry and are used in the classification of algebraic surfaces

Rational surfaces

• A rational surface is an algebraic surface that is birational to the projective plane
• The minimal models of rational surfaces are:
  • The projective plane (κ = -∞)
  • The Hirzebruch surfaces (κ = -∞)
• Every rational surface can be obtained from the projective plane or a Hirzebruch surface by a sequence of blowups
  • Blowing up a point on a surface replaces the point with a copy of the projective line (an exceptional curve)
• Rational surfaces have a rich geometry and are used in the classification of algebraic surfaces
  • For example, cubic surfaces and del Pezzo surfaces are rational surfaces obtained by blowing up points in the projective plane

Enriques-Kodaira classification of surfaces

Overview of the classification

• The Enriques-Kodaira classification is a complete classification of algebraic surfaces up to birational equivalence, based on their Kodaira dimension and other invariants
• The classification consists of 10 classes:
  1. Rational surfaces (κ = -∞)
  2. Ruled surfaces (κ = -∞ or 1)
  3. Abelian surfaces (κ = 0)
  4. K3 surfaces (κ = 0)
  5. Enriques surfaces (κ = 0)
  6. Bielliptic surfaces (κ = 0)
  7. Elliptic surfaces (κ = 1)
  8. Surfaces of general type (κ = 2)
  9. Surfaces of Kodaira dimension 1 (not elliptic)
  10. Surfaces of Kodaira dimension 0 (not abelian, K3, Enriques, or bielliptic)
• Each class is characterized by specific properties of the canonical divisor, the plurigenera, and the fundamental group

Relationship to other classification schemes

• The classification is obtained by studying the minimal models of algebraic surfaces and their birational transformations
  • Minimal models are surfaces that cannot be simplified further by contracting exceptional curves
• The classification is closely related to the classification of complex analytic surfaces and the classification of compact complex manifolds of dimension 2
  • Every algebraic surface is a complex analytic surface, but not every complex analytic surface is algebraic
• The classification has been generalized to higher dimensions (e.g., the minimal model program for 3-folds and the Iitaka fibration for varieties of arbitrary dimension)
  • However, the classification becomes more complicated in higher dimensions, and many questions remain open