9.2 Genus and Riemann-Roch theorem

Genus and the Riemann-Roch theorem are key concepts in understanding algebraic curves. They help measure a curve's complexity and provide tools for analyzing its properties. These ideas are crucial for classifying curves and studying their behavior.

The genus relates to a curve's "holes" and degree, while Riemann-Roch connects meromorphic functions to divisors. Together, they offer powerful insights into curve structure and form the basis for deeper explorations in algebraic geometry.

Genus of Algebraic Curves

Definition and Geometric Interpretation

Top images from around the web for Definition and Geometric Interpretation

• 

  Quadric Surfaces · Calculus View original

  Is this image relevant?

• 

  Euler Spirals, Curves, Spiro View original

  Is this image relevant?

• 

  Number theory - Wikipedia View original

  Is this image relevant?

• 

  Quadric Surfaces · Calculus View original

  Is this image relevant?

• 

  Euler Spirals, Curves, Spiro View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition and Geometric Interpretation

• 

  Quadric Surfaces · Calculus View original

  Is this image relevant?

• 

  Euler Spirals, Curves, Spiro View original

  Is this image relevant?

• 

  Number theory - Wikipedia View original

  Is this image relevant?

• 

  Quadric Surfaces · Calculus View original

  Is this image relevant?

• 

  Euler Spirals, Curves, Spiro View original

  Is this image relevant?

1 of 3

• The genus of an algebraic curve is a non-negative integer that measures the complexity or "holes" in the curve
• Geometrically, the genus represents the maximum number of non-intersecting simple closed curves that can be drawn on the surface without separating it
• The genus of a smooth, projective, irreducible algebraic curve over an algebraically closed field is a birational invariant
• The genus of a curve is related to its topological properties, such as its Euler characteristic and Betti numbers

Computing the Genus of Plane Curves

• For a smooth, projective, irreducible curve C of degree d in the complex projective plane, the genus g is given by the formula: $g = (d-1)(d-2)/2$
• This formula shows that the genus of a plane curve increases quadratically with its degree
• Curves of genus 0 are called rational curves and have a parametrization by rational functions (lines and conics)
• Curves of genus 1 are called elliptic curves and have a rich geometric and algebraic structure (smooth, irreducible plane curves of degree 3)

Riemann-Roch Theorem for Curves

Statement and Implications

• The Riemann-Roch theorem is a fundamental result in algebraic geometry that relates the dimension of the space of meromorphic functions on a curve to its genus and divisors
• For a smooth, projective, irreducible curve C over an algebraically closed field, and a divisor D on C, the Riemann-Roch theorem states: $l(D) - l(K-D) = deg(D) - g + 1$, where l(D) is the dimension of the space of meromorphic functions on C with poles bounded by D, K is the canonical divisor, and g is the genus of C
• The theorem provides a powerful tool for computing the dimension of linear systems on curves and understanding the behavior of meromorphic functions

Applications and Consequences

• The Riemann-Roch theorem has numerous applications in the study of algebraic curves, including the classification of curves, the computation of the gonality, and the study of special divisors such as Weierstrass points
• The theorem can be used to compute the dimension of the space of meromorphic functions on a curve with prescribed pole orders at specific points
• The Riemann-Roch theorem is a key ingredient in the proof of the Riemann-Hurwitz formula, which relates the genera of curves in a covering map

Dimension of Meromorphic Functions

Computing Dimensions using Riemann-Roch

• The dimension of the space of meromorphic functions on a curve C with poles bounded by a divisor D, denoted l(D), can be computed using the Riemann-Roch theorem
• For a canonical divisor K and a divisor D on C, the dimension l(D) is given by the formula: $l(D) = deg(D) - g + 1 + l(K-D)$, where g is the genus of the curve
• In the case of the trivial divisor (D = 0), the dimension l(0) equals 1, corresponding to the constant functions on the curve

Special Cases and Vanishing

• For a divisor D with degree greater than or equal to 2g-1, where g is the genus, the dimension l(D) is given by $l(D) = deg(D) - g + 1$, as the term l(K-D) vanishes
• The computation of l(D) provides information about the number of linearly independent meromorphic functions on the curve with prescribed pole orders at specific points
• The vanishing of l(K-D) for divisors of high degree is related to the Kodaira vanishing theorem in higher dimensions

Genus vs Degree of Plane Curves

Relationship and Classification

• For a smooth, projective, irreducible plane curve C of degree d in the complex projective plane, the genus g and the degree d are related by the formula: $g = (d-1)(d-2)/2$
• This formula demonstrates that the genus of a plane curve increases quadratically with its degree
• The relationship between genus and degree provides a way to classify and study algebraic curves based on their intrinsic geometric properties
• As the degree of a plane curve increases, so does its genus, leading to more complex geometric and algebraic properties

Examples and Special Cases

• Curves of genus 0 are called rational curves and have a parametrization by rational functions. The only smooth, irreducible plane curves of genus 0 are lines (d=1) and conics (d=2)
• Curves of genus 1 are called elliptic curves and have a rich geometric and algebraic structure. Smooth, irreducible plane curves of genus 1 have degree 3
• Hyperelliptic curves are curves of genus g that admit a degree 2 map to the projective line. They can be described by equations of the form $y^2 = f(x)$, where f(x) is a polynomial of degree $2g+1$ or $2g+2$
• Plane curves of high degree and genus, such as curves of degree 4 (genus 3) and degree 5 (genus 6), exhibit increasingly complex geometric and algebraic properties, and are the subject of active research in algebraic geometry