🌿Computational Algebraic Geometry Unit 8 – Rational Maps & Birational Geometry

Key Concepts and Definitions

• Rational maps defined as functions between algebraic varieties that can be expressed as ratios of polynomials
• Birational equivalence occurs when two varieties are isomorphic outside a subvariety of codimension at least 2
• Regular functions are polynomial functions on an affine variety
• Rational functions are quotients of regular functions
  • Can have poles and be undefined at certain points
• Morphisms are maps between varieties that are defined by polynomials
  • Preserve the algebraic structure of the varieties
• Isomorphisms are bijective morphisms with an inverse that is also a morphism
• Blow-ups resolve singularities by replacing a point with a projective space of lines through that point

Rational Maps: Basics and Properties

• Rational maps can be represented using homogeneous coordinates
  • Allows for a consistent representation of points at infinity
• Composition of rational maps is also a rational map
  • Enables the study of dynamics and iteration of rational functions
• Rational maps are not necessarily defined everywhere
  • Points where the denominator vanishes are called base points or indeterminacy points
• Degree of a rational map is the degree of the polynomials used to define it
  • Determines the number of preimages of a generic point
• Rational maps can be extended to projective varieties
  • Allows for a more complete and consistent theory
• Rational maps are dominant if their image is dense in the codomain
• Birational maps are rational maps with a rational inverse
  • Establish an equivalence between varieties up to birational isomorphism

Birational Geometry Fundamentals

• Birational geometry studies properties of varieties that are invariant under birational equivalence
• Minimal models are varieties that are smooth and whose canonical divisor is nef (numerically effective)
  • Unique up to isomorphism within a birational equivalence class
• Canonical divisor is a divisor associated with the top differential form on a smooth variety
  • Encodes information about the geometry and topology of the variety
• Kodaira dimension measures the growth rate of sections of multiples of the canonical divisor
  • Birational invariant that characterizes the geometry of a variety
• Minimal Model Program (MMP) is an approach to classifying algebraic varieties up to birational equivalence
  • Involves a sequence of birational transformations and contractions to obtain a minimal model
• Fano varieties are varieties whose anticanonical divisor is ample
  • Play a central role in the classification of algebraic varieties
• Rationally connected varieties are varieties where any two points can be connected by a rational curve
  • Form a special class of varieties with rich geometric properties

Computational Techniques for Rational Maps

• Gröbner bases can be used to compute the implicit equations of a rational map
  • Allows for the study of the image and fibers of the map
• Resultants and discriminants can detect the presence of base points and singularities
• Elimination theory techniques help in computing the closure of the image of a rational map
• Numerical algebraic geometry methods (homotopy continuation) can solve systems of polynomial equations arising from rational maps
  • Enables the computation of preimages and fibers
• Toric varieties and their maps can be studied using combinatorial techniques
  • Exploits the connection between geometry and convex polytopes
• Intersection theory algorithms (Schubert calculus) can determine the degree of a rational map
• Symbolic computation software (Macaulay2, Singular, Sage) provides tools for working with rational maps and algebraic varieties

Applications in Algebraic Geometry

• Rational maps are used to construct new varieties from existing ones
  • Blow-ups, projections, and birational modifications
• Parametrization of curves and surfaces using rational functions
  • Enables the study of their geometry and arithmetic
• Resolution of singularities using sequences of blow-ups
  • Produces smooth models of singular varieties
• Study of the birational geometry of moduli spaces
  • Classifies geometric objects up to equivalence
• Cremona transformations are birational self-maps of projective spaces
  • Used in the study of birational rigidity and finite subgroups of the Cremona group
• Rational curves on varieties provide information about their geometry and arithmetic
  • Rationally connected varieties and the Minimal Model Program
• Arithmetic dynamics studies the behavior of rational maps over number fields and finite fields
  • Relates to questions in number theory and cryptography

Advanced Topics and Theorems

• Theorem of resolution of singularities (Hironaka) guarantees the existence of a smooth birational model for any variety over a field of characteristic zero
• Minimal Model Program (MMP) conjectures aim to establish the existence of minimal models and provide a classification of algebraic varieties
  • Mori fiber spaces and the abundance conjecture
• Sarkisov program studies the factorization of birational maps between Mori fiber spaces
  • Relates to the structure of the Cremona group
• Kontsevich's formula expresses the number of rational curves on a variety in terms of Gromov-Witten invariants
  • Connects birational geometry with enumerative geometry and mirror symmetry
• Batyrev-Borisov mirror symmetry constructs mirror pairs of Calabi-Yau varieties using reflexive polytopes and toric geometry
• Kawamata-Viehweg vanishing theorem is a powerful tool in the study of the positivity of divisors and the geometry of varieties
• Minimal Model Program with scaling is a refined version of the MMP that incorporates the canonical divisor and provides a more efficient algorithm

Practical Examples and Problem-Solving

• Computing the birational type of a given algebraic variety
  • Determining if it is rational, stably rational, or of general type
• Finding explicit birational maps between varieties
  • Constructing inverses and studying their properties
• Resolving the singularities of a curve or surface
  • Computing the blow-up at a singular point and its effect on the geometry
• Determining the Kodaira dimension of a projective variety
  • Using the properties of the canonical divisor and pluricanonical maps
• Studying the birational geometry of moduli spaces
  • Classifying algebraic curves, surfaces, or higher-dimensional varieties up to birational equivalence
• Computing the degree and indeterminacy locus of a rational map
  • Using resultants, Gröbner bases, and elimination theory
• Applying the Minimal Model Program to a specific class of varieties
  • Constructing minimal models and Mori fiber spaces for Fano or Calabi-Yau varieties

Connections to Other Areas of Mathematics

• Birational geometry is closely related to complex geometry and the classification of complex manifolds
• Rational maps and birational transformations appear in dynamical systems and the study of iterated functions
  • Julia sets and the Fatou conjecture
• Diophantine geometry uses rational points and rational curves to study the arithmetic of varieties
  • Mordell conjecture and Faltings' theorem
• Toric geometry provides a combinatorial approach to the study of rational maps and birational geometry
  • Toric varieties, polytopes, and fans
• Gromov-Witten theory and mirror symmetry relate birational geometry to the enumerative geometry of curves and the topology of Calabi-Yau manifolds
• Birational anabelian geometry studies the relationship between the birational geometry of a variety and its étale fundamental group
  • Section conjecture and the Grothendieck-Serre anabelian conjectures
• Birational geometry over finite fields and number fields has applications in coding theory and cryptography
  • Rational points, curves, and surfaces over finite fields