7.2 Dimension of varieties and coordinate rings

Dimension is a crucial concept in algebraic geometry, measuring the "size" of varieties. It's defined as the Krull dimension of a variety's coordinate ring, which is the length of the longest chain of prime ideals.

Understanding dimension helps us classify varieties and study their properties. It's related to tangent spaces, behaves predictably under morphisms, and plays a key role in analyzing curves, surfaces, and higher-dimensional varieties.

Dimension of varieties

Determining dimension using coordinate rings

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• The dimension of an affine or projective variety X is defined as the Krull dimension of its coordinate ring A(X)
  • For an affine variety X, the coordinate ring A(X) is the quotient ring k[x₁, ..., xₙ]/I(X), where I(X) is the ideal of polynomials vanishing on X
  • For a projective variety X, the coordinate ring A(X) is the homogeneous coordinate ring k[x₀, ..., xₙ]/I(X), where I(X) is the homogeneous ideal of polynomials vanishing on X
• The Krull dimension of a ring R is the supremum of the lengths of all chains of prime ideals in R
  • To determine the dimension of a variety, one computes the Krull dimension of its coordinate ring by finding the longest chain of prime ideals
• The dimension of an irreducible variety is equal to the transcendence degree of its function field over the base field k

Examples of determining dimension

• Consider the affine variety X defined by the equation x² + y² = 1 in A². The coordinate ring A(X) is k[x, y]/(x² + y² - 1), and the longest chain of prime ideals is (0) ⊂ (x - 1, y), so dim(X) = 1
• For the projective variety Y defined by the homogeneous equation x³ + y³ + z³ = 0 in ℙ², the coordinate ring A(Y) is k[x, y, z]/(x³ + y³ + z³), and the longest chain of prime ideals is (0) ⊂ (x, y), so dim(Y) = 1
• The affine variety Z defined by the equations x² + y² = 1 and z = 0 in A³ has coordinate ring k[x, y, z]/(x² + y² - 1, z), and the longest chain of prime ideals is (0) ⊂ (x - 1, y, z), so dim(Z) = 0

Dimension and coordinate ring

Relating dimension to Krull dimension

• The dimension of an affine or projective variety X is equal to the Krull dimension of its coordinate ring A(X)
  • The Krull dimension of a ring R is the supremum of the lengths of all chains of prime ideals in R
  • For an affine variety X, the coordinate ring A(X) is the quotient ring k[x₁, ..., xₙ]/I(X), where I(X) is the ideal of polynomials vanishing on X
  • For a projective variety X, the coordinate ring A(X) is the homogeneous coordinate ring k[x₀, ..., xₙ]/I(X), where I(X) is the homogeneous ideal of polynomials vanishing on X
• The dimension of an irreducible variety is equal to the transcendence degree of its function field over the base field k

Dimension and tangent spaces

• The dimension of a variety X is also equal to the dimension of the tangent space at any smooth point of X
  • The tangent space at a point x ∈ X is the vector space of derivations of the local ring O_{X,x}
  • For an affine variety X ⊂ Aⁿ, the tangent space at a smooth point x is isomorphic to the kernel of the Jacobian matrix of the defining equations evaluated at x
• The dimension of the tangent space at a singular point may be larger than the dimension of the variety

Dimension under morphisms

Behavior of dimension under morphisms

• A morphism of varieties f: X → Y is a continuous map that is locally given by polynomials
• The dimension of the image of a morphism f: X → Y is always less than or equal to the dimension of X
  • If f: X → Y is a surjective morphism, then dim(X) ≥ dim(Y). If f is also finite, then dim(X) = dim(Y)
• The dimension of the fibers of a morphism can vary, but for a flat morphism, the dimension of the fibers is constant

Fibrations and dimension

• A fibration is a morphism f: X → Y such that for every point y ∈ Y, the fiber f⁻¹(y) is a variety of constant dimension
  • For a fibration f: X → Y, the dimension formula holds: dim(X) = dim(Y) + dim(f⁻¹(y)), where y is any point in Y
• Examples of fibrations include projections of product varieties (X × Y → Y) and bundle maps (E → B, where E is a vector bundle over B)
• The dimension of the total space of a fibration is the sum of the dimensions of the base and the fiber

Dimension for geometric study

Dimension of specific varieties

• A curve is a variety of dimension 1. Examples include lines, conics, and elliptic curves
  • The genus of a smooth projective curve C is related to its dimension by the formula dim H⁰(C, Ω^1) = g, where Ω^1 is the sheaf of differential forms on C
• A surface is a variety of dimension 2. Examples include planes, quadrics, and K3 surfaces
  • The Hodge diamond of a smooth projective surface encodes its Hodge numbers, which are related to its dimension and geometric properties
• A hypersurface is a variety defined by a single polynomial equation. The dimension of a hypersurface in ℙⁿ is n-1
  • The degree of a hypersurface is related to its dimension and the degree of its defining polynomial

Dimension and singularities

• The dimension of the singular locus of a variety is always strictly less than the dimension of the variety itself
  • For example, the singular locus of a curve consists of isolated points (dimension 0), while the singular locus of a surface can contain curves (dimension 1)
• The dimension of the intersection of two varieties X and Y in a smooth ambient space is related to their individual dimensions by the formula dim(X ∩ Y) ≥ dim(X) + dim(Y) - dim(ambient space), with equality holding if the intersection is transverse
  • Transverse intersections occur when the tangent spaces of X and Y span the tangent space of the ambient variety at every point of intersection