Glossary

2.1 Definition and examples of affine varieties

4 min read•july 30, 2024

Affine varieties are the building blocks of algebraic geometry, defined by polynomial equations over algebraically closed fields. They represent geometric objects like lines, curves, and surfaces, forming a bridge between algebra and geometry.

Understanding affine varieties is crucial for grasping more complex concepts in algebraic geometry. We'll explore their definition, examples, and key properties, including the and the correspondence between varieties and ideals.

Affine varieties and algebraic sets

Definition of affine varieties

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  • An is the set of solutions of a system of polynomial equations in nn variables over an algebraically closed field kk
  • The algebraic set of an affine variety VV is the set of all points (x1,…,xn)(x_1, \ldots, x_n) in knk^n that satisfy the polynomial equations defining VV
    • For example, the algebraic set of the affine variety defined by the equation x2+y2=1x^2 + y^2 = 1 over the field of complex numbers is the unit circle in the complex plane
  • An affine variety VV is irreducible if it cannot be written as the union of two proper subvarieties
    • Equivalently, the I(V)I(V) is a prime ideal

Zariski topology and ideals

  • The ideal I(V)I(V) of an affine variety VV is the set of all polynomials f(x1,…,xn)f(x_1, \ldots, x_n) that vanish on VV, meaning f(a1,…,an)=0f(a_1, \ldots, a_n) = 0 for all points (a1,…,an)(a_1, \ldots, a_n) in VV
    • For instance, the ideal of the affine variety defined by x2+y2=1x^2 + y^2 = 1 is the ideal generated by the polynomial x2+y2−1x^2 + y^2 - 1
  • The Zariski topology on knk^n is defined by taking the closed sets to be the affine varieties (algebraic sets)
    • In this topology, the closure of a set is the smallest affine variety containing it
    • The Zariski topology is coarser than the usual Euclidean topology on knk^n

Examples of affine varieties

Affine varieties in low dimensions

  • In the affine plane ( 2), examples of affine varieties include lines, parabolas, ellipses, and hyperbolas
    • The line y=mx+by = mx + b is an affine variety defined by the linear equation y−mx−b=0y - mx - b = 0
    • The parabola y=x2y = x^2 is an affine variety defined by the quadratic equation y−x2=0y - x^2 = 0
  • In 3-dimensional affine space, examples include planes, spheres, ellipsoids, and cubic surfaces
    • The plane ax+by+cz+d=0ax + by + cz + d = 0 is an affine variety defined by a linear equation
    • The sphere x2+y2+z2=r2x^2 + y^2 + z^2 = r^2 is an affine variety defined by a quadratic equation

Trivial and compound affine varieties

  • The empty set and the entire affine space knk^n are both trivial examples of affine varieties in any dimension nn
  • The intersection of two affine varieties is always an affine variety
    • For example, the intersection of the plane z=0z = 0 and the sphere x2+y2+z2=1x^2 + y^2 + z^2 = 1 is the unit circle in the xyxy-plane
  • The union of two affine varieties is an affine variety if and only if one is contained in the other

Ideals and affine varieties

Correspondence between affine varieties and radical ideals

  • There is a one-to-one correspondence (bijection) between affine varieties in knk^n and radical ideals in the polynomial ring k[x1,…,xn]k[x_1, \ldots, x_n]
    • Given an affine variety VV, its ideal I(V)I(V) is always a radical ideal
    • Conversely, given any radical ideal II in k[x1,…,xn]k[x_1, \ldots, x_n], the set V(I)V(I) of points where all polynomials in II vanish is an affine variety
  • The correspondence V↦I(V)V \mapsto I(V) and I↦V(I)I \mapsto V(I) are inclusion-reversing
    • If V1⊆V2V_1 \subseteq V_2 then I(V2)⊆I(V1)I(V_2) \subseteq I(V_1)
    • If I1⊆I2I_1 \subseteq I_2 then V(I2)⊆V(I1)V(I_2) \subseteq V(I_1)

Hilbert's Nullstellensatz

  • states that if kk is algebraically closed, the correspondence between affine varieties and radical ideals is a bijection
    • In other words, every radical ideal is the ideal of some affine variety, and every affine variety is the zero set of some radical ideal
    • This fundamental theorem establishes a deep connection between algebraic geometry and commutative algebra

Dimension and degree of affine varieties

Dimension of affine varieties

  • The dimension of an affine variety VV is the Krull dimension of its k[V]=k[x1,…,xn]/I(V)k[V] = k[x_1, \ldots, x_n] / I(V), which is the supremum of the lengths of chains of prime ideals in k[V]k[V]
    • Intuitively, the dimension measures the number of independent directions in which VV extends
  • The dimension of VV is equal to the number of independent variables in a set of equations defining VV, after eliminating redundant variables
    • For example, the parabola y=x2y = x^2 has dimension 1, since it can be described by a single independent variable xx

Degree of affine varieties

  • If VV is an affine variety of dimension dd in knk^n, then almost all linear subspaces of knk^n of codimension dd intersect VV in a finite number of points, called the degree of VV
    • The degree measures the size or complexity of VV
    • For instance, a line has degree 1, a quadratic curve has degree 2, and a cubic surface has degree 3
  • The degree of VV is equal to the leading coefficient of the Hilbert polynomial of k[V]k[V], which describes the dimension of the vector space of polynomials on VV of a given degree
    • The Hilbert polynomial encodes important information about the structure of VV and its coordinate ring

Key Terms to Review (16)

Affine Variety: An affine variety is a subset of affine space defined as the common zero set of a collection of polynomials. These varieties are fundamental objects in algebraic geometry, connecting geometric concepts with algebraic expressions through their coordinate rings and properties.
Closed Set: A closed set is a fundamental concept in topology, defined as a set that contains all its limit points. In the context of affine varieties, closed sets are critical because they correspond to the solution sets of polynomial equations, which describe the geometric shapes we study. Understanding closed sets helps to identify the properties of affine varieties, including their irreducibility and dimensionality.
Coordinate Ring: The coordinate ring of an affine variety is a way to represent the algebraic structure of the variety through polynomials. Specifically, it consists of all polynomial functions defined on the affine space corresponding to the variety, allowing for a bridge between geometric objects and algebraic expressions. This concept is crucial in understanding properties such as dimension, ideals, and relationships between different types of varieties.
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.
Embedding: An embedding is a mathematical mapping that allows one space to be represented within another, preserving the structure and properties of the original space. This concept is crucial in various areas of algebraic geometry, as it establishes relationships between different varieties, allowing us to study them in a unified way. By embedding affine varieties into projective spaces, for example, we can analyze their properties and behaviors in a broader context.
Hilbert's Nullstellensatz: Hilbert's Nullstellensatz is a fundamental theorem in algebraic geometry that establishes a connection between ideals in polynomial rings and the geometric properties of algebraic varieties. It essentially states that there is a correspondence between the radical of an ideal and the points of the affine variety it defines, linking algebraic expressions to their geometric counterparts.
Homomorphism: A homomorphism is a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces, that respects the operations defined on those structures. This means that if you apply the operation in the first structure and then map it to the second, you get the same result as if you mapped each element first and then applied the operation in the second structure. Homomorphisms help to connect different algebraic systems and show how they can behave similarly under certain operations.
Ideal: An ideal is a special subset of a ring that is closed under addition and under multiplication by any element of the ring. This concept is essential in algebraic geometry, as ideals can be used to define algebraic varieties and play a key role in understanding the relationships between geometric objects and algebraic equations.
Irreducible Affine Variety: An irreducible affine variety is a type of algebraic set that cannot be expressed as the union of two smaller non-empty algebraic sets. In other words, it is a variety that cannot be decomposed into simpler components. This concept is fundamental in understanding the structure of affine varieties and their geometric properties, as it emphasizes the notion of 'connectedness' in the realm of algebraic geometry.
Isomorphism: An isomorphism is a mathematical mapping that establishes a one-to-one correspondence between two structures, showing that they are essentially the same in terms of their properties and operations. This concept connects various algebraic structures and geometric entities, revealing how different representations can share the same underlying characteristics, such as affine varieties and projective varieties.
Morphism: A morphism is a structure-preserving map between two mathematical objects, typically within the context of algebraic geometry where it connects varieties. This concept allows for the exploration of relationships between different varieties, examining how their structures relate and translate through functions and coordinates.
Reduced Affine Variety: A reduced affine variety is an algebraic set that does not contain any nilpotent elements in its coordinate ring, which means it has no repeated roots in its defining polynomials. This property ensures that the structure of the variety is as 'simple' as possible, allowing for clearer geometric interpretations and the application of various algebraic techniques. Reduced affine varieties are fundamental in the study of algebraic geometry as they represent the most basic form of geometric objects, where each point corresponds uniquely to a maximal ideal in the coordinate ring.
Spectrum of a Ring: The spectrum of a ring, denoted as Spec(R), is the set of all prime ideals of the ring R, equipped with the Zariski topology. This concept links algebra and geometry by allowing us to treat the prime ideals of a ring as geometric points, which correspond to the affine varieties associated with the ring. The spectrum provides a way to study the properties of rings through their prime ideals and forms the basis for understanding affine varieties in algebraic geometry.
The variety of a polynomial: The variety of a polynomial is the set of all points in affine space that satisfy the polynomial equation. This concept is fundamental in algebraic geometry as it connects algebraic expressions to geometric shapes, allowing us to visualize the solutions of polynomial equations as geometric objects in space.
Variety Corresponding to an Ideal: A variety corresponding to an ideal is the set of all points in affine space that satisfy a given polynomial ideal. This concept links algebra and geometry by showing how algebraic properties of polynomials can describe geometric shapes, allowing us to represent solutions to systems of polynomial equations geometrically as varieties.
Zariski topology: Zariski topology is a specific type of topology defined on the set of points in affine and projective spaces, focusing on the algebraic properties of varieties. This topology is uniquely defined by the concept of closed sets being algebraic sets, which are defined by polynomial equations. It connects algebra and geometry, emphasizing the relationship between solutions to polynomial equations and geometric structures.
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