🕴🏼Elementary Algebraic Geometry Unit 5 – Morphisms and Rational Maps

Key Concepts and Definitions

• Morphisms are structure-preserving maps between algebraic varieties that respect the underlying algebraic structure
• Varieties are geometric objects defined by polynomial equations in affine or projective space
• Affine varieties are defined by polynomial equations in affine space $\mathbb{A}^n$, while projective varieties are defined in projective space $\mathbb{P}^n$
• Regular functions are polynomial functions defined on an open subset of a variety
• Rational functions are quotients of regular functions, i.e., functions of the form $\frac{f}{g}$ where $f$ and $g$ are regular functions and $g \neq 0$
• Rational maps are maps between varieties defined by rational functions
  • They are not necessarily defined everywhere, as the denominator may vanish at some points
• Isomorphisms are bijective morphisms with an inverse that is also a morphism
  • They preserve the algebraic structure and establish an equivalence between varieties

Types of Morphisms

• Regular morphisms (or morphisms) are maps between varieties that are defined by polynomial functions on each affine open set
  • They are continuous in the Zariski topology and respect the algebraic structure of the varieties
• Closed immersions are morphisms that identify the domain as a closed subvariety of the codomain
  • They are injective and have a closed image
• Open immersions are morphisms that identify the domain as an open subvariety of the codomain
  • They are injective and have an open image
• Finite morphisms are morphisms with finite fibers, i.e., the preimage of each point is a finite set
• Dominant morphisms are morphisms with dense image, i.e., the closure of the image is equal to the codomain
• Birational morphisms are dominant morphisms that admit a rational inverse
  • They establish a birational equivalence between varieties

Properties of Morphisms

• Morphisms are continuous in the Zariski topology, which is defined by taking closed sets to be the zero sets of polynomial equations
• Composition of morphisms is again a morphism, allowing for the study of categories of varieties and morphisms
• Morphisms induce pullback maps on the rings of regular functions, preserving the algebraic structure
• The preimage of a closed set under a morphism is closed, and the preimage of an open set is open
• Morphisms between affine varieties can be described by homomorphisms between their coordinate rings
  • This establishes a correspondence between geometric and algebraic properties
• Morphisms between projective varieties can be described by homogeneous polynomial maps that respect the equivalence relation defining projective space
• The graph of a morphism is a closed subvariety of the product of the domain and codomain

Rational Maps: Basics and Examples

• Rational maps are maps between varieties that are defined by rational functions, i.e., quotients of regular functions
• Unlike morphisms, rational maps are not necessarily defined everywhere, as the denominator may vanish at some points
• The domain of definition of a rational map is an open dense subset of the domain variety
• Rational maps can be extended to morphisms by resolving the indeterminacies, i.e., finding a variety on which the map is defined everywhere
  • This process is called regularization or resolution of singularities
• Examples of rational maps include:
  • The map from the affine line $\mathbb{A}^1$ to itself given by $x \mapsto \frac{1}{x}$, which is undefined at $x = 0$
  • The map from the projective plane $\mathbb{P}^2$ to itself given by $[x:y:z] \mapsto [yz:xz:xy]$, which is defined everywhere
  • The map from the projective line $\mathbb{P}^1$ to the affine plane $\mathbb{A}^2$ given by $[x:y] \mapsto (\frac{x}{y}, \frac{y}{x})$, which is undefined at $[0:1]$ and $[1:0]$

Comparing Morphisms and Rational Maps

• Morphisms are more restrictive than rational maps, as they must be defined everywhere and respect the algebraic structure
• Every morphism is a rational map, but not every rational map is a morphism
• Rational maps can be extended to morphisms by resolving the indeterminacies, but this may require changing the domain variety
• Morphisms are better suited for studying the global structure of varieties, while rational maps are more flexible and can be used to study birational geometry
• Birational maps, which are rational maps admitting a rational inverse, establish a weaker equivalence between varieties than isomorphisms
  • Varieties that are birationally equivalent may have different global structure but share many local properties
• The study of rational maps and birational geometry is a central theme in algebraic geometry, with applications to classification problems and the minimal model program

Applications in Algebraic Geometry

• Morphisms and rational maps are fundamental tools in the study of algebraic varieties and their properties
• They allow for the comparison and classification of varieties based on their intrinsic geometry
• Morphisms can be used to study the moduli spaces of algebraic varieties, which parametrize families of varieties with certain properties
  • These moduli spaces are themselves algebraic varieties and can be studied using morphisms
• Rational maps are used in the birational classification of varieties, which aims to identify varieties up to birational equivalence
  • The minimal model program seeks to find canonical representatives for birational equivalence classes of varieties
• Morphisms and rational maps also play a role in the study of algebraic cycles, intersection theory, and cohomology theories for varieties
• In arithmetic geometry, morphisms and rational maps are used to study the arithmetic properties of varieties defined over number fields or finite fields
  • The study of rational points and Diophantine equations is a central theme in this area

Common Challenges and Misconceptions

• Understanding the distinction between morphisms and rational maps, and when each is appropriate to use
• Recognizing that not all maps between varieties are morphisms or rational maps, and that some may not respect the algebraic structure
• Dealing with indeterminacies and singularities when working with rational maps
  • Resolving indeterminacies may require blowing up the variety or changing the domain
• Distinguishing between local and global properties of morphisms and rational maps
  • Some properties (e.g., injectivity, surjectivity) may hold locally but not globally
• Navigating the various notions of equivalence between varieties (e.g., isomorphism, birational equivalence) and understanding their relationships
• Working with varieties and maps defined over fields other than the complex numbers, where the geometry may behave differently
• Applying the abstract concepts and machinery of morphisms and rational maps to concrete examples and computations

Practice Problems and Exercises

• Determine whether a given map between affine or projective varieties is a morphism, a rational map, or neither
• Compute the domain of definition for a given rational map and identify the points of indeterminacy
• Find the preimage of a closed or open set under a morphism
• Determine whether a morphism is an isomorphism, closed immersion, open immersion, finite, or dominant
• Compute the pullback map on rings of regular functions induced by a morphism
• Extend a rational map to a morphism by resolving indeterminacies
• Find a birational map between two given varieties and compute the rational inverse
• Determine whether two varieties are birationally equivalent and find a birational map between them if possible
• Compute the graph of a morphism or rational map as a subvariety of the product
• Use morphisms and rational maps to study the geometry and arithmetic of specific examples of varieties (e.g., curves, surfaces, hypersurfaces)