1.4 Dimension and degree of varieties

Dimension and degree are crucial concepts in algebraic geometry, helping us understand the size and complexity of varieties. Dimension measures the "number of directions" in a variety, while degree quantifies its intersection with linear subspaces.

These concepts bridge geometry and algebra, allowing us to study varieties using both approaches. Understanding dimension and degree is essential for analyzing intersections, singularities, and other properties of algebraic varieties.

Dimension of a variety

Defining dimension for irreducible and reducible varieties

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• The dimension of an irreducible variety is the transcendence degree of its function field over the base field
• For a reducible variety, the dimension is the maximum of the dimensions of its irreducible components
• The empty set has a dimension of -1, while a point has a dimension of 0
• A variety's dimension is independent of the ambient space in which it is embedded

Properties and theorems related to dimension

• The dimension of a variety equals the dimension of its tangent space at any smooth point
• The Intersection Theorem states that the dimension of the intersection of two varieties is at least the sum of their dimensions minus the dimension of the ambient space
  • This theorem provides a lower bound for the dimension of the intersection of two varieties
  • For example, if two curves (dimension 1) intersect in a plane (dimension 2), the dimension of their intersection is at least 1 + 1 - 2 = 0, which means they intersect in points (dimension 0)

Degree of projective varieties

Defining degree for irreducible and reducible projective varieties

• The degree of an irreducible projective variety is the number of points in its intersection with a general linear subspace of complementary dimension
  • For example, the degree of a curve (dimension 1) in projective space is the number of points in which it intersects a general hyperplane (codimension 1)
• For a reducible projective variety, the degree is the sum of the degrees of its irreducible components
• The degree measures the complexity or "size" of a projective variety

Properties and theorems related to degree

• The degree of a hypersurface in projective space equals the degree of its defining polynomial
  • A hypersurface is a variety defined by a single polynomial equation
  • For instance, a quadric surface in projective 3-space, defined by a degree 2 polynomial, has degree 2
• Bézout's Theorem: The number of points (counting multiplicities) in the intersection of two projective varieties equals the product of their degrees, provided the intersection is finite
  • This theorem is useful for computing the number of intersection points between two varieties
  • For example, a line (degree 1) and a conic curve (degree 2) in the projective plane intersect in 1 × 2 = 2 points (counting multiplicities)

Dimension and degree of varieties

Examples of computing dimension and degree

• Affine n-space ($A^n$) has dimension $n$, and its degree is not defined
• Projective n-space ($P^n$) has dimension $n$ and degree 1
• A hypersurface in $A^n$ or $P^n$ defined by a non-constant polynomial has dimension one less than the ambient space
  • For example, a curve in the affine or projective plane has dimension 1
• The degree of a hypersurface in $P^n$ defined by a polynomial of degree $d$ is $d$
  • A plane curve defined by a degree 3 polynomial has degree 3
• The graph of a polynomial function from $A^n$ to $A^m$ has dimension equal to $n$
  • The graph of a polynomial function from the affine line to the affine plane has dimension 1
• The Segre embedding of $P^n × P^m$ in $P^((n+1)(m+1)-1)$ has dimension $n + m$
  • The Segre embedding of $P^1 × P^2$ in $P^5$ has dimension 1 + 2 = 3

Dimension vs transcendence degree

Relating dimension and transcendence degree

• The transcendence degree of a field extension $K/k$ is the maximal number of algebraically independent elements in $K$ over $k$
• For an irreducible affine variety $V$, the dimension of $V$ equals the transcendence degree of its coordinate ring $k[V]$ over $k$
• For an irreducible projective variety $V$, the dimension of $V$ equals the transcendence degree of its homogeneous coordinate ring $k[V]$ over $k$ minus 1

Connecting geometry and algebra

• The dimension of a variety can be computed by finding the transcendence degree of its function field over the base field
  • This provides a way to determine the dimension of a variety using algebraic methods
• The concepts of dimension and transcendence degree link the geometric and algebraic aspects of varieties
  • They allow us to study geometric properties of varieties using algebraic tools, and vice versa
  • For example, the dimension of a variety can be understood both geometrically (the "number of independent directions" in the variety) and algebraically (the transcendence degree of its function field)