7.2 Projective varieties and their properties

Projective varieties are the zero sets of homogeneous polynomials in projective space. They're closed under projective transformations and have properties like dimension, irreducibility, and smoothness that help us understand their geometry.

Projective varieties extend affine varieties to include points at infinity. This makes them compact and gives us powerful tools like Bézout's theorem for studying intersections. Understanding projective varieties is key to modern algebraic geometry.

Properties of Projective Varieties

Definition and Basic Properties

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• Projective varieties are defined as the zero locus of a set of homogeneous polynomials in projective space
  • The zero locus is the set of points where all the polynomials vanish simultaneously
  • Homogeneous polynomials have the same degree in each term (e.g., $x^2y + xyz + z^3$)
• Projective varieties are closed under projective transformations, which are invertible linear transformations of the homogeneous coordinates
  • Projective transformations preserve the degree and homogeneity of the defining polynomials
  • Examples of projective transformations include rotations, translations, and scaling in projective space
• The dimension of a projective variety is the maximum number of algebraically independent homogeneous polynomials that vanish on the variety
  • Algebraic independence means that no polynomial can be expressed as a polynomial combination of the others
  • The dimension is also equal to the dimension of the tangent space at a smooth point on the variety

Irreducibility and Smoothness

• Projective varieties are irreducible if they cannot be expressed as the union of two proper subvarieties
  • Proper subvarieties have lower dimension than the original variety
  • Irreducibility is a fundamental property that simplifies the study of varieties
• Smooth projective varieties are those without singular points, meaning the Jacobian matrix of the defining polynomials has full rank at every point on the variety
  • The Jacobian matrix contains the partial derivatives of the defining polynomials
  • Singular points are where the tangent space has lower dimension than expected (e.g., cusps, nodes, or self-intersections)
• The degree of a projective variety is the number of intersection points with a generic linear subspace of complementary dimension
  • The degree measures the complexity of the variety and is invariant under projective transformations
  • For example, a plane curve of degree $d$ intersects a generic line in $d$ points (counted with multiplicity)

Projective vs Affine Varieties

Definitions and Relationships

• Affine varieties are defined by polynomial equations in affine space, while projective varieties are defined by homogeneous polynomial equations in projective space
  • Affine space is a vector space without a fixed origin, while projective space adds points at infinity
  • Polynomial equations in affine space have arbitrary degree terms (e.g., $x^2 + y - 1 = 0$)
• Projective space can be viewed as the union of affine spaces glued together along their hyperplanes at infinity, with projective varieties extending affine varieties to include points at infinity
  • The hyperplane at infinity consists of the points where the last homogeneous coordinate vanishes
  • Projective varieties can have multiple affine patches, each corresponding to a different hyperplane at infinity
• Affine varieties can be obtained from projective varieties by dehomogenizing the defining equations with respect to one of the homogeneous coordinates
  • Dehomogenization sets one of the homogeneous coordinates to 1 and eliminates it from the equations
  • The resulting equations define an affine variety in the remaining coordinates
• Projective varieties can be obtained from affine varieties by homogenizing the defining equations and considering the closure in projective space
  • Homogenization introduces a new variable to make all terms have the same degree
  • The closure adds limit points at infinity to the affine variety

Invariant Properties

• Projective varieties are compact, meaning they are closed and bounded in the Euclidean topology, while affine varieties are not necessarily compact
  • Compactness is a topological property that is preserved under continuous maps
  • Non-compact affine varieties can have points that "escape to infinity" in some direction
• Some geometric properties, such as degree and genus, are invariant under the correspondence between affine and projective varieties
  • The degree of an affine variety is equal to the degree of its projective closure
  • The genus of a smooth curve is a topological invariant that measures the number of "holes" in the curve
  • For example, a circle has genus 0, while a torus has genus 1

Intersection of Projective Varieties

Basic Properties and Bézout's Theorem

• The intersection of two projective varieties is again a projective variety, defined by the union of the defining equations of the original varieties
  • The intersection inherits the properties of projective varieties, such as closure under projective transformations
  • The intersection may be reducible even if the original varieties are irreducible
• Bézout's theorem states that the degree of the intersection of two projective varieties is equal to the product of their degrees, counting multiplicities
  • Multiplicities account for the presence of multiple intersection points or higher-dimensional components
  • Bézout's theorem is a fundamental result in enumerative algebraic geometry
• The multiplicity of an intersection point is the local degree of the intersection at that point, which can be computed using the Hilbert function or the multiplicity of the local ring
  • The Hilbert function measures the growth of the dimension of the space of homogeneous polynomials of a given degree
  • The multiplicity of the local ring is the degree of the projectivized tangent cone at the point

Proper and Excess Intersections

• The intersection of two varieties is proper if their codimensions add up to the codimension of the ambient space, and the intersection has the expected dimension
  • The codimension is the difference between the dimension of the ambient space and the dimension of the variety
  • Proper intersections are the "generic" case and satisfy Bézout's theorem
• Excess intersection occurs when the codimensions do not add up as expected, often due to the presence of an irreducible component in the intersection with higher than expected dimension
  • Excess intersections can be resolved by perturbing the varieties or by considering the intersection in a higher-dimensional space
  • The excess intersection formula relates the actual intersection to the expected intersection via the Chern classes of the normal bundles
• The Euler characteristic of the intersection of two varieties can be computed using the Euler characteristics of the original varieties and their intersection, via the inclusion-exclusion principle
  • The Euler characteristic is a topological invariant that measures the alternating sum of the Betti numbers
  • The inclusion-exclusion principle relates the cardinality of the union of sets to the cardinalities of the individual sets and their intersections

Projective Techniques for Algebraic Geometry

Curves and Surfaces

• Projective methods can be used to study the geometry of algebraic curves, such as determining their genus, singularities, and rational points
  • The genus of a curve can be computed using the degree-genus formula or the Riemann-Roch theorem
  • Singularities of curves can be resolved by blowing up the curve at the singular points
  • Rational points on curves can be found using the method of descent or the Mordell-Weil theorem
• The projective plane can be used to analyze the intersection of algebraic curves, including the computation of intersection multiplicities and the resolution of singularities
  • Intersection multiplicities can be computed using the resultant or the Puiseux expansion
  • Singularities of plane curves can be resolved by a sequence of blow-ups, leading to the minimal model of the curve
• Projective techniques can simplify the study of algebraic surfaces by considering their projective embeddings and analyzing their hyperplane sections
  • Projective embeddings realize a surface as a subvariety of projective space, often via the complete linear system of a divisor
  • Hyperplane sections are the intersections of the surface with hyperplanes in the ambient projective space
  • The geometry of the surface can be studied via the properties of its hyperplane sections (e.g., the canonical class, the intersection form)

Higher Dimensions and Applications

• The projective space can be used to study the geometry of higher-dimensional varieties, such as the classification of algebraic threefolds and the study of Calabi-Yau manifolds
  • The minimal model program aims to classify algebraic varieties up to birational equivalence by finding their minimal models
  • Calabi-Yau manifolds are complex Kähler manifolds with trivial canonical class, which play a central role in string theory and mirror symmetry
• Projective methods can be applied to solve enumerative problems in algebraic geometry, such as counting the number of curves or surfaces satisfying certain conditions
  • Enumerative problems often involve the intersection theory of the moduli spaces of curves or surfaces
  • The Gromov-Witten invariants and the Donaldson-Thomas invariants are important enumerative invariants that count curves on varieties
• Projective techniques can be used in computational algebraic geometry to efficiently represent and manipulate algebraic varieties, such as in the computation of Gröbner bases and the resolution of singularities
  • Gröbner bases are special generating sets of ideals that provide a way to solve systems of polynomial equations and to study the geometry of varieties
  • Resolution of singularities is the process of finding a smooth variety that is birationally equivalent to a given singular variety
  • Projective methods can be used to simplify the computation and to provide explicit resolutions in many cases (e.g., toric varieties, determinantal varieties)