7.3 Dimension theory for projective varieties

Dimension theory for projective varieties builds on concepts from affine varieties, extending them to the projective setting. It explores how to measure the "size" of these geometric objects, connecting ideas from algebra and geometry.

This topic dives into the relationships between projective varieties, their affine cones, and homogeneous coordinate rings. It also covers how dimensions behave under projective morphisms and rational maps, tying everything together in a cohesive framework.

Dimension Theory for Projective Varieties

Extending Dimension Theory to Projective Varieties and Graded Rings

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• The dimension of a projective variety X is defined as the dimension of any affine open subset U of X, which is equal to the dimension of the affine variety U as an affine variety
• For a graded ring S, the dimension of Proj(S) is equal to the Krull dimension of S minus 1
• The dimension of a projective variety X is equal to the transcendence degree of its function field over the base field
  • The function field of X is the field of rational functions on X, which is the quotient field of any affine open subset U of X
  • The transcendence degree measures the number of algebraically independent elements needed to generate the function field over the base field
• The dimension of a projective variety X is also equal to the maximal length of a chain of irreducible closed subvarieties of X
  • A chain of subvarieties is a sequence of subvarieties X_0 ⊂ X_1 ⊂ ... ⊂ X_n = X, where each X_i is a proper subvariety of X_{i+1}
  • The length of the chain is the number of strict inclusions in the sequence
• The dimension of a projective variety X is invariant under projective isomorphisms
  • A projective isomorphism is a morphism f: X -> Y between projective varieties that has an inverse morphism g: Y -> X such that f ∘ g and g ∘ f are identity morphisms

Dimension of Projective Varieties vs Affine Cones

Relationship Between Projective Varieties and Their Affine Cones

• For a projective variety X in P^n, its affine cone C(X) is the affine variety in A^{n+1} defined by the same homogeneous polynomials that define X
  • The affine cone C(X) is obtained by considering the homogeneous coordinates of X as affine coordinates in A^{n+1}
  • The origin of A^{n+1} is always included in C(X)
• The dimension of the affine cone C(X) is equal to the dimension of the projective variety X plus 1
  • This is because the affine cone has one additional dimension corresponding to the scaling of the homogeneous coordinates
• The projective variety X can be recovered from its affine cone C(X) by taking the quotient of C(X) minus the origin by the action of the multiplicative group of the base field
  • The multiplicative group acts by scaling the homogeneous coordinates, which corresponds to the equivalence relation defining the projective space
• The affine cone C(X) is irreducible if and only if the projective variety X is irreducible
  • Irreducibility means that the variety cannot be written as the union of two proper closed subvarieties
  • The correspondence between X and C(X) preserves irreducibility

Dimension Computation in Homogeneous Rings

Using Homogeneous Coordinate Rings to Compute Dimension

• The homogeneous coordinate ring of a projective variety X in P^n is the graded ring S = k[x_0, ..., x_n]/I, where I is the homogeneous ideal defining X
  • The variables x_0, ..., x_n correspond to the homogeneous coordinates of P^n
  • The ideal I consists of all homogeneous polynomials that vanish on X
• The dimension of X is equal to the Krull dimension of S minus 1
  • The Krull dimension of a ring is the maximal length of a chain of prime ideals in the ring
  • The minus 1 accounts for the grading of S, which adds an extra dimension
• The Krull dimension of S can be computed using the maximal length of a chain of prime ideals in S
  • A prime ideal is an ideal P such that if ab ∈ P, then either a ∈ P or b ∈ P
  • The chain of prime ideals corresponds to a chain of irreducible closed subvarieties of X
• The dimension of X can also be computed using the Hilbert polynomial of S, which encodes information about the dimensions of the graded pieces of S
  • The Hilbert polynomial is a polynomial P(t) such that P(n) equals the dimension of the n-th graded piece of S for sufficiently large n
  • The degree of the Hilbert polynomial equals the dimension of X

Hilbert Functions and Polynomials

• The Hilbert function of a graded ring S is the function H(n) that gives the dimension of the n-th graded piece of S
  • The n-th graded piece of S consists of all homogeneous elements of S of degree n
  • The dimension is taken as a vector space over the base field
• The Hilbert polynomial P(t) of S is a polynomial such that P(n) = H(n) for sufficiently large n
  • The existence of the Hilbert polynomial follows from the fact that H(n) agrees with a polynomial for large n
  • The Hilbert polynomial encodes asymptotic information about the growth of the graded pieces of S
• The degree of the Hilbert polynomial P(t) equals the dimension of Proj(S)
  • This provides a way to compute the dimension of a projective variety from its homogeneous coordinate ring
  • The leading coefficient of P(t) is related to the degree of the projective variety

Dimension Under Projective Morphisms

Projective Morphisms and Fiber Dimensions

• A projective morphism f: X -> Y between projective varieties is a morphism that can be represented by homogeneous polynomials of the same degree in the homogeneous coordinate rings of X and Y
  • The homogeneous polynomials defining f must be compatible with the grading of the coordinate rings
  • Projective morphisms are well-defined on the level of projective varieties, independent of the choice of homogeneous coordinates
• The dimension of the fibers of a projective morphism f: X -> Y is upper semicontinuous, meaning that for any d, the set {y in Y | dim(f^{-1}(y)) >= d} is closed in Y
  • The fiber of f over a point y ∈ Y is the preimage f^{-1}(y), which is a projective subvariety of X
  • Upper semicontinuity implies that the dimension of the fibers can only jump up on closed subsets of Y
• If f: X -> Y is a projective morphism between irreducible projective varieties, then dim(X) >= dim(Y), and equality holds if and only if f is surjective and the general fiber of f is finite
  • Surjectivity means that every point of Y is in the image of f
  • The general fiber refers to the fiber over a general point, meaning a point in a dense open subset of Y
  • Finiteness of the general fiber means that it consists of a finite number of points

Dimension and Dominant Rational Maps

• A rational map f: X ---> Y between projective varieties is a morphism defined on an open dense subset U of X
  • Rational maps allow for maps that are not defined everywhere, but are defined on a dense open subset
  • The closure of the image of f is a closed subvariety of Y
• The dimension of the closure of the image of f is at most the dimension of X
  • This follows from the fact that f is defined on an open dense subset of X, and the image of f is contained in Y
• If f: X ---> Y is a dominant rational map between irreducible projective varieties (meaning the image of f is dense in Y), then dim(X) >= dim(Y), and equality holds if and only if the general fiber of f is finite
  • Dominance implies that the image of f is dense in Y, so the closure of the image is equal to Y
  • The general fiber of f refers to the fiber over a general point in the open dense subset where f is defined
  • Finiteness of the general fiber means that it consists of a finite number of points
  • The inequality dim(X) >= dim(Y) follows from the fact that f is defined on an open dense subset of X, and the general fiber has dimension zero