10.2 Ruled surfaces and rational surfaces

Ruled surfaces are like mathematical playgrounds, where every point hangs out on a straight line. They're the cool kids of geometry, showing up as cylinders, cones, and even fancy hyperboloids. These surfaces come in different flavors, from developable to skew, each with its own unique twist.

Rational surfaces take things up a notch, using fancy rational functions to describe themselves. They're like the VIP club of surfaces, including the projective plane and Hirzebruch surfaces. While all rational surfaces are ruled, not all ruled surfaces make the cut as rational.

Ruled Surfaces and Properties

Definition and Parametrization

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• A ruled surface is a surface that can be swept out by a moving straight line, called a ruling or generator
• Every point on a ruled surface lies on a straight line that is contained in the surface
• Ruled surfaces can be parametrized by a curve and a family of lines, with the curve determining the direction of the ruling at each point
• Examples of ruled surfaces include:
  • Cylinders
  • Cones
  • Hyperboloids of one sheet
  • Hyperbolic paraboloids

Degree, Class, and Classification

• The degree of a ruled surface is the degree of the equation defining the surface, which is related to the number of rulings passing through a general point in space
• The class of a ruled surface is the degree of the dual surface, which is the set of all tangent planes to the surface
• Ruled surfaces can be classified as:
  • Developable surfaces: Tangent plane is constant along each ruling (cylinders and cones)
  • Non-developable (skew) surfaces: Tangent plane varies along each ruling (hyperboloids and hyperbolic paraboloids)

Rational vs Ruled Surfaces

Rational Surfaces

• A rational surface is a surface that can be parametrized by rational functions, i.e., the ratio of two polynomials in two variables
• Examples of rational surfaces:
  • Projective plane ($\mathbb{P}^2$)
  • Hirzebruch surfaces ($\mathbb{F}_n$)
• Rational surfaces can be obtained by:
  • Blowing up points on the projective plane
  • Blowing down exceptional curves on a surface
• The Picard number of a rational surface, which measures the rank of its Picard group (the group of divisors modulo linear equivalence), is related to the number of blowups and blowdowns performed

Relationship between Rational and Ruled Surfaces

• Every rational surface is a ruled surface, but not every ruled surface is rational
• The minimal model of a rational surface is the simplest surface birationally equivalent to it, obtained by contracting all exceptional curves

Birational Geometry of Rational Surfaces

Birational Equivalence and Maps

• Two surfaces are birationally equivalent if there exist rational maps between them that are inverses of each other outside of a lower-dimensional subset
• Birational maps preserve many geometric properties, such as:
  • Degree
  • Arithmetic genus
  • Kodaira dimension of a surface
• The birational geometry of rational surfaces is determined by the minimal model and the configuration of exceptional curves

Cremona Group and Factorization

• The Cremona group is the group of birational automorphisms of the projective plane, which is generated by:
  • Linear automorphisms
  • Standard quadratic transformations
• Birational maps between rational surfaces can be factored into a sequence of blowups and blowdowns, as described by the Castelnuovo-Enriques-Noether theorem

Minimal Ruled Surfaces

Definition and Properties

• A minimal ruled surface is a ruled surface that does not contain any exceptional curves, i.e., curves with self-intersection number -1
• Every ruled surface is birationally equivalent to a minimal ruled surface, obtained by contracting all exceptional curves
• The Picard number of a minimal ruled surface is 2, and its Picard group is generated by:
  • The class of a fiber
  • The class of a section

Hirzebruch Surfaces

• Minimal rational ruled surfaces are isomorphic to Hirzebruch surfaces, which are $\mathbb{P}^1$-bundles over $\mathbb{P}^1$
• The Hirzebruch surface $\mathbb{F}_n$ is defined by the projectivization of the vector bundle $\mathcal{O}(0) \oplus \mathcal{O}(n)$ over $\mathbb{P}^1$, where $n$ is a non-negative integer
• The self-intersection number of the minimal section (the section with the smallest self-intersection) on a Hirzebruch surface $\mathbb{F}_n$ is $-n$