3.2 Projective varieties and their properties

Projective varieties are the geometric objects at the heart of algebraic geometry. They're defined as zero sets of homogeneous polynomials in projective space, giving us a powerful way to study geometric shapes using algebraic tools.

These varieties have fascinating properties like dimension, degree, and singularities. We can break them down into irreducible pieces, study their intersections, and even define a special ring structure called the Chow ring to capture their geometry.

Projective varieties and ideals

Definition and ideal-theoretic description

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• A projective variety is a subset of projective space that is the zero locus of a collection of homogeneous polynomials
• The ideal generated by the homogeneous polynomials that vanish on a projective variety is called the defining ideal of the variety
• The projective coordinate ring of a projective variety is the quotient ring of the polynomial ring by the defining ideal of the variety

Topology and irreducibility

• The Zariski topology on projective space is defined by taking projective varieties as the closed sets
• A projective variety is irreducible if it cannot be written as the union of two proper subvarieties (i.e., it is not the union of two smaller projective varieties)

Properties of projective varieties

Dimension, degree, and singularities

• The dimension of a projective variety is the maximum length of a chain of irreducible subvarieties, minus one
• The degree of a projective variety is the number of points in its intersection with a general linear subspace of complementary dimension (i.e., a linear subspace whose dimension plus the dimension of the variety equals the dimension of the ambient projective space)
• A point on a projective variety is singular if the Jacobian matrix of the defining polynomials has rank less than the codimension of the variety at that point
  • The codimension is the difference between the dimension of the ambient projective space and the dimension of the variety
• The singular locus of a projective variety is the set of all singular points, which forms a subvariety of lower dimension
• The tangent space at a smooth point of a projective variety is the kernel of the Jacobian matrix at that point

Decomposition into irreducible components

• Every projective variety can be uniquely decomposed as a finite union of irreducible components
• The irreducible components of a projective variety correspond to the minimal prime ideals of its defining ideal
• The dimension of a reducible projective variety is the maximum of the dimensions of its irreducible components

Irreducibility vs Reducibility

Definitions and examples

• A projective variety is irreducible if it cannot be written as the union of two proper subvarieties
  • Example: the projective curve defined by the equation $y^2z = x^3 - xz^2$ (an elliptic curve) is irreducible
• A projective variety is reducible if it can be written as the union of two proper subvarieties
  • Example: the projective curve defined by the equation $xy = 0$ is reducible, as it is the union of the lines $x = 0$ and $y = 0$

Relationship to prime ideals

• The irreducible components of a projective variety correspond to the minimal prime ideals of its defining ideal
  • A prime ideal is an ideal $P$ such that if $ab \in P$, then either $a \in P$ or $b \in P$
  • A minimal prime ideal is a prime ideal that does not contain any smaller prime ideals

Intersection of projective varieties

Bézout's theorem and intersection multiplicity

• Bézout's theorem states that the intersection of two projective varieties of complementary dimension consists of a finite number of points, counted with multiplicity
• The multiplicity of a point in the intersection of two projective varieties can be computed using the Hilbert polynomial of the local ring at that point
  • The Hilbert polynomial measures the growth of the dimensions of the homogeneous components of the local ring
• The intersection multiplicity satisfies certain axioms, such as being additive on unions and multiplicative on products

Intersection product and Chow ring

• The intersection product of two subvarieties of a projective variety is a cycle that represents their set-theoretic intersection with multiplicities
  • A cycle is a formal linear combination of subvarieties with integer coefficients
• The Chow ring of a projective variety is the ring generated by the classes of subvarieties modulo rational equivalence, with the intersection product as multiplication
  • Two cycles are rationally equivalent if they can be connected by a family of cycles parametrized by a rational curve