Glossary

10.2 Ruled surfaces and rational surfaces

3 min read•july 30, 2024

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 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 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 (P2\mathbb{P}^2)
    • Hirzebruch surfaces (Fn\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
    • Kodaira 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 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
    • The class of a

Hirzebruch Surfaces

  • Minimal rational ruled surfaces are isomorphic to Hirzebruch surfaces, which are P1\mathbb{P}^1-bundles over P1\mathbb{P}^1
  • The Hirzebruch surface Fn\mathbb{F}_n is defined by the projectivization of the vector bundle O(0)⊕O(n)\mathcal{O}(0) \oplus \mathcal{O}(n) over P1\mathbb{P}^1, where nn is a non-negative integer
  • The self-intersection number of the minimal section (the section with the smallest self-intersection) on a Hirzebruch surface Fn\mathbb{F}_n is −n-n

Key Terms to Review (16)

Birational Equivalence: Birational equivalence is a concept in algebraic geometry where two varieties are considered equivalent if they can be related by rational maps that are inverses of each other on dense open subsets. This idea connects to regular functions and morphisms, as birational maps allow for understanding the behavior of functions across varieties. It's crucial for studying properties like dimension and classification, especially when dealing with surfaces and their geometric properties.
Cremona transformations: Cremona transformations are rational maps between projective spaces that can be understood as a way to modify or 'transform' geometric objects, particularly in the context of algebraic geometry. These transformations allow for the study of properties of varieties, especially rational surfaces and ruled surfaces, by providing a means to relate different geometrical configurations through a birational map. They can be visualized as operations that take points in one projective space and map them to another, revealing deeper insights into the structure of algebraic varieties.
David Mumford: David Mumford is a prominent mathematician known for his significant contributions to algebraic geometry, particularly in the areas of rational maps, birational equivalence, and the development of modern algebraic geometry techniques. His work emphasizes the geometric aspects of algebraic structures and has had a lasting impact on the understanding of surfaces, schemes, and dimensions within this mathematical field.
Dimension: Dimension is a fundamental concept in geometry and algebra that refers to the number of independent directions or parameters needed to describe a space or object. In algebraic geometry, it helps classify varieties based on their geometric properties, influencing how they are represented and understood in terms of both affine and projective spaces.
Fano's Theorem: Fano's Theorem states that if a projective variety has a rational point, then it is covered by rational curves. This theorem highlights the relationship between rationality and the existence of certain types of curves on algebraic varieties, emphasizing the significance of ruled surfaces in understanding rational surfaces.
Fiber: In algebraic geometry, a fiber refers to the preimage of a point under a continuous map, typically from a space such as a variety to another space. Fibers are essential in understanding the geometric structure and properties of surfaces, allowing mathematicians to analyze how different points relate to each other and the overall shape of the object.
Fibration: A fibration is a structure that describes a map between two topological spaces, where the pre-image of every point in the target space is a space called a fiber. This concept plays a critical role in understanding how varieties are connected through maps, especially in the context of studying their dimensions and properties. In addition to its importance in algebraic geometry, fibrations can illustrate how specific geometric structures, such as ruled and rational surfaces, relate to each other.
Genus: Genus refers to a topological invariant that measures the number of 'holes' in a surface or curve. In algebraic geometry, genus helps classify curves and surfaces based on their geometric properties, revealing important information about their structure and behavior. It connects various concepts such as singularities, intersection theory, and the classification of surfaces.
Hirzebruch's Theorem: Hirzebruch's Theorem is a significant result in algebraic geometry that relates to the properties of complex projective varieties, specifically in describing the characteristics of ruled surfaces and rational surfaces. This theorem highlights how the topology of these surfaces can be determined using intersection numbers, providing tools for understanding their geometric and algebraic structure. It connects deep ideas from algebraic topology with the classification of algebraic varieties, particularly focusing on the role of the Hirzebruch classes in this classification.
John Harris: John Harris is a mathematician known for his contributions to algebraic geometry, particularly in the context of ruled and rational surfaces. His work often emphasizes the interplay between geometric structures and their algebraic properties, helping to deepen the understanding of how these surfaces behave under various conditions and transformations.
Minimal model program: The minimal model program is a framework in algebraic geometry aimed at classifying algebraic varieties by simplifying them to their 'minimal' forms. This approach focuses on transforming varieties through a series of operations, such as blow-ups and contractions, to obtain a simpler model that retains essential geometric properties. By working towards minimal models, this program connects deeply with the classification of algebraic surfaces and provides insights into specific types of surfaces, such as ruled and rational ones.
Projective Space: Projective space is a mathematical construct that extends the concept of Euclidean space by introducing a notion of points at infinity, allowing for the study of geometric properties that remain invariant under projection. This framework is crucial for understanding various properties of projective varieties, including their irreducibility, and helps establish connections between algebraic structures and geometric intuition.
Rational Surface: A rational surface is a type of algebraic surface that can be parametrized by rational functions. This means that there exists a dominant morphism from the projective plane to the surface, indicating that the surface behaves like a 'rational' object in algebraic geometry. Rational surfaces are crucial in understanding the classification of algebraic surfaces and are closely related to ruled surfaces, as many rational surfaces can be constructed as bundles over curves.
Ruled surface: A ruled surface is a type of surface that can be generated by moving a straight line through space, meaning every point on the surface can be reached by a line segment. Ruled surfaces are important in the classification of algebraic surfaces as they serve as a bridge between different types of surfaces, particularly rational surfaces and non-rational ones. They exhibit unique geometric properties, allowing for interesting visualizations and analyses in algebraic geometry.
Section: In algebraic geometry, a section refers to a global choice of a point in the fibers of a morphism, essentially providing a way to select an object over each point in the base space. This concept is key when discussing properties of ruled surfaces and rational surfaces, as well as elliptic surfaces and K3 surfaces, since sections help in understanding the structure and behavior of these geometric objects.
Self-intersection: Self-intersection refers to the phenomenon where a geometric object intersects itself at one or more points. This concept is particularly relevant in the study of algebraic varieties, where self-intersections can reveal important information about the structure and properties of the object. Understanding self-intersections is crucial for analyzing ruled and rational surfaces, as they can indicate singularities and play a role in determining the surface's geometric properties.
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