Algebraic sets are the building blocks of algebraic geometry, representing solutions to polynomial equations. They bridge algebra and geometry, allowing us to study geometric objects through their defining equations and vice versa.
In this part of the chapter, we'll explore how algebraic sets are defined, their properties, and their geometric interpretations. We'll see how these concepts lay the foundation for understanding more complex ideas in algebraic geometry.
Algebraic sets and their properties
Definition and basic properties
- An algebraic set is the solution set of a system of polynomial equations over a field
- It is a subset of affine or projective space defined by the vanishing of a collection of polynomials
- The Zariski topology on affine or projective space is defined by taking algebraic sets as the closed sets
- This topology is coarser than the Euclidean topology
Irreducibility and dimension
- An algebraic set is irreducible if it cannot be written as the union of two proper algebraic subsets
- Irreducible algebraic sets are the building blocks of algebraic geometry
- The dimension of an algebraic set is the maximum length of a chain of irreducible subsets
- Length is defined as the number of strict inclusions
- The dimension of an irreducible algebraic set equals the Krull dimension of its coordinate ring
- The singular locus of an algebraic set is the subset of points where the Jacobian matrix of the defining equations does not have full rank
- The complement of the singular locus is called the smooth locus
- Algebraic sets can be classified and studied based on properties such as reducibility, singularity, and dimension
Geometric interpretation of algebraic sets
Visualization in various dimensions
- Algebraic sets in affine or projective space can be visualized as geometric objects
- The geometry of an algebraic set reflects the properties of its defining equations
- In the affine plane (dimension 2), algebraic sets are curves defined by polynomial equations in two variables
- Examples include lines, parabolas, ellipses, hyperbolas, and more complicated curves
- In affine 3-space, algebraic sets are surfaces defined by polynomial equations in three variables
- Examples include planes, spheres, cylinders, and other surfaces that can be described algebraically
- In higher dimensions, algebraic sets can represent hypersurfaces, curves, or other geometric objects
- Visualization becomes more challenging, but techniques from algebraic geometry still apply
Singularities and smooth points
- Singular points of an algebraic set correspond to geometric singularities
- Examples include cusps, nodes, or self-intersections
- The smooth locus represents the non-singular part of the geometric object
- Techniques such as projections, intersections, and parametrizations can be used to study the geometric properties of algebraic sets
- These techniques relate the algebraic descriptions to their geometric counterparts
Algebraic sets vs defining equations
Ideals and coordinate rings
- An algebraic set is defined by a collection of polynomial equations
- The structure of these equations determines the geometric properties of the algebraic set
- The ideal of an algebraic set is the set of all polynomials that vanish on the set
- This ideal captures the algebraic relations satisfied by the points of the set
- The coordinate ring of an algebraic set is the quotient of the polynomial ring by the ideal of the set
- It encodes the algebraic functions on the set and reflects its geometric properties
Nullstellensatz and operations on algebraic sets
- The Nullstellensatz establishes a correspondence between radical ideals and algebraic sets
- Every radical ideal is the ideal of some algebraic set, and conversely, the ideal of an algebraic set is always radical
- Operations on algebraic sets correspond to operations on their defining ideals
- Union corresponds to sum, intersection to intersection, and complement to quotient
- The prime decomposition of an ideal corresponds to the irreducible decomposition of the corresponding algebraic set
- This allows studying reducible algebraic sets in terms of their irreducible components
Construction and manipulation of algebraic sets
Constructing algebraic sets
- Constructing algebraic sets involves finding polynomial equations that define the desired geometric object
- This can be done by using algebraic techniques or by geometric reasoning
- Parametrization is a technique for describing algebraic sets using rational functions
- It allows representing the points of an algebraic set in terms of a smaller number of parameters
- Elimination theory provides methods for eliminating variables from a system of polynomial equations
- This can be used to project an algebraic set onto a lower-dimensional space or to compute the intersection of algebraic sets
- The resultant and discriminant are algebraic tools for studying the common solutions of polynomial equations
- They can be used to determine the existence and multiplicity of intersection points
- Gröbner bases are a powerful computational tool for solving systems of polynomial equations
- They provide a systematic way to manipulate and simplify the defining equations of an algebraic set
- Algebraic sets can be transformed using maps between affine or projective spaces
- These maps can be defined by polynomials and can be used to study the relationships between different algebraic sets
- Real algebraic sets are algebraic sets defined over the real numbers
- They have additional geometric properties and can be studied using techniques from real algebraic geometry
- Examples include the real Nullstellensatz and semi-algebraic sets