🕴🏼Elementary Algebraic Geometry Unit 7 – Dimension Theory

Key Concepts and Definitions

• Dimension measures the size or complexity of an algebraic variety or scheme
  • Intuitively corresponds to the number of independent parameters needed to describe points
• Irreducible components are the basic building blocks of algebraic varieties
  • Cannot be written as the union of two proper closed subvarieties
• Codimension is the difference between the dimension of the ambient space and the dimension of a subvariety
• Hilbert polynomial encodes information about the dimension and degree of a projective variety
  • Leading coefficient relates to the degree and dimension
• Transcendence degree extends the concept of dimension to fields
  • Number of algebraically independent elements needed to generate the field over a base field
• Dimension of the tangent space at a point reflects the local dimension of the variety near that point
• Regular local rings correspond to non-singular points on a variety
  • Their dimension equals the dimension of the variety at that point

Historical Context and Development

• Study of dimension in algebraic geometry has roots in classical projective geometry and invariant theory
• Hilbert's work on invariants and bases led to the development of Hilbert polynomials
  • Used to study dimension and degree of projective varieties
• Krull introduced the concept of dimension for commutative rings in the 1920s
  • Extended to dimension of irreducible varieties
• Zariski's work on algebraic varieties in the 1940s and 1950s further developed dimension theory
  • Introduced the concept of codimension and studied local properties
• Grothendieck's reformulation of algebraic geometry in terms of schemes in the 1960s
  • Provided a unified framework for studying dimension
• Modern developments include the study of singular varieties and the use of homological methods
  • Intersection theory and Chow groups relate to the study of dimension

Dimension in Algebraic Geometry

• Dimension is a fundamental invariant of algebraic varieties and schemes
• For an irreducible variety, the dimension is the transcendence degree of its function field over the base field
  • Equals the Krull dimension of its coordinate ring
• Dimension of a reducible variety is the maximum dimension of its irreducible components
• Dimension is preserved under birational equivalence
  • Varieties with isomorphic function fields have the same dimension
• Dimension behaves well under morphisms
  • For a surjective morphism, the dimension of the target is at most the dimension of the source
• Fiber dimension theorem relates the dimensions of the fibers and the target of a morphism
  • For a morphism $f: X \to Y$, $\dim X = \dim Y + \dim f^{-1}(y)$ for a general point $y \in Y$
• Dimension can be computed using the Hilbert polynomial for projective varieties
  • The degree of the Hilbert polynomial equals the dimension

Krull Dimension

• Krull dimension is a notion of dimension for commutative rings and schemes
• For a commutative ring $R$, the Krull dimension is the supremum of lengths of chains of prime ideals
  • $\dim R = \sup\{n \mid \exists P_0 \subsetneq P_1 \subsetneq \cdots \subsetneq P_n, P_i \text{ prime}\}$
• Krull dimension agrees with the geometric notion of dimension for affine varieties
  • For an affine variety $X = \operatorname{Spec} A$, $\dim X = \dim A$
• Krull dimension is preserved under localization
  • For a multiplicative subset $S \subset R$, $\dim R = \dim S^{-1}R$
• Krull's principal ideal theorem bounds the codimension of a principal ideal
  • If $R$ is a Noetherian ring and $f \in R$ is a non-zero-divisor, then $\dim R/(f) \geq \dim R - 1$
• Krull dimension can be computed using the height of prime ideals
  • For a Noetherian ring $R$, $\dim R = \max\{\operatorname{ht} P \mid P \text{ prime ideal of } R\}$

Geometric Interpretation

• Dimension corresponds to the number of independent parameters needed to describe points on a variety
  • For curves (dimension 1), one parameter (often denoted $t$) is sufficient
  • For surfaces (dimension 2), two parameters $(u,v)$ are needed
• Dimension can be visualized in terms of the intersection of hypersurfaces
  • The intersection of $n$ hypersurfaces in $\mathbb{P}^n$ is generically a finite set of points (dimension 0)
  • The intersection of $n-1$ hypersurfaces in $\mathbb{P}^n$ is generically a curve (dimension 1)
• Singular points on a variety can be thought of as points where the local dimension is higher than expected
  • For example, a node on a curve locally looks like the intersection of two lines (dimension 0)
• Dimension is related to the degree of freedom in deforming a variety
  • Higher-dimensional varieties have more ways to be deformed or modified
• Fiber dimension theorem has a geometric interpretation
  • The dimension of a variety is the sum of the dimension of a general fiber and the dimension of the base

Applications in Algebraic Varieties

• Dimension is used to classify algebraic varieties and understand their properties
  • Curves (dimension 1), surfaces (dimension 2), threefolds (dimension 3), etc.
• Dimension is a key ingredient in the statement of the Riemann-Roch theorem for curves
  • Relates the dimension of the space of global sections of a line bundle to its degree
• Kodaira dimension measures the size of the canonical bundle of a variety
  • Used to classify varieties in birational geometry
• Dimension appears in the definition of the Hilbert scheme parametrizing subschemes of a fixed variety
  • The Hilbert scheme has a stratification by the dimension of the subschemes
• Dimension is used to formulate intersection theory on varieties
  • The intersection product of subvarieties has dimension equal to the sum of their codimensions
• Dimension is related to the singularities of a variety
  • The dimension of the singular locus is strictly smaller than the dimension of the variety

Computational Techniques

• Gröbner basis methods can be used to compute the dimension of affine varieties
  • The dimension is the number of variables minus the number of elements in a Gröbner basis
• Hilbert series of a graded ring encodes information about its dimension
  • The dimension is the order of the pole at $t=1$
• Singular value decomposition (SVD) can be used to estimate the dimension of a variety from a sample of points
  • The number of significant singular values indicates the dimension
• Numerical algebraic geometry techniques (e.g., homotopy continuation) can compute dimensions of varieties
  • By tracking the solutions of a parameterized system of equations
• Dimension can be computed using the Hilbert polynomial for projective varieties
  • The degree of the Hilbert polynomial equals the dimension
• Combinatorial methods (e.g., simplicial complexes) can be used to study the dimension of toric varieties
  • The dimension equals the dimension of the corresponding polytope

Advanced Topics and Extensions

• Intersection theory studies the intersections of subvarieties and their relation to dimension
  • Chow groups and rings provide a framework for intersection theory
• Motivic integration extends integration to the setting of algebraic varieties and schemes
  • Involves measuring the dimension of the space of arcs on a variety
• Hodge theory studies the structure of the cohomology of complex algebraic varieties
  • Hodge numbers and the Hodge diamond encode dimensional information
• Berkovich spaces are analogues of algebraic varieties over non-Archimedean fields
  • Dimension theory can be developed in this setting
• Tropical geometry studies degenerations of algebraic varieties to polyhedral complexes
  • The dimension of a tropical variety is related to the dimension of the original variety
• Derived algebraic geometry incorporates higher categorical structures into the study of varieties
  • Dimension theory can be extended to derived schemes and stacks
• Characteristic $p$ methods, such as the Frobenius morphism and $p$-adic cohomology
  • Provide tools for studying dimension in positive characteristic