The Riemann-Roch theorem is a powerful tool in algebraic geometry, connecting line bundles, divisors, and cohomology. It relates the dimension of global sections to the degree and genus of curves, or to the intersection numbers and canonical divisor for surfaces.

This theorem has far-reaching applications in studying linear systems, special divisors, and geometric properties of curves and surfaces. It's crucial for understanding canonical embeddings, Kodaira dimensions, and the existence of special curves on surfaces.

Riemann-Roch Theorem for Curves and Surfaces

Statement of the Theorem

• The Riemann-Roch theorem relates the dimension of the space of global sections of a line bundle to its degree and the genus of the curve or surface
• For a smooth projective curve $C$ of genus $g$ and a divisor $D$ on $C$, the Riemann-Roch theorem states that $dim H⁰(C, O(D)) - dim H¹(C, O(D)) = deg(D) - g + 1$
• For a smooth projective surface $S$ and a divisor $D$ on $S$, the Riemann-Roch theorem states that $χ(S, O(D)) = 1/2 D·(D - K) + χ(S, O)$, where $K$ is the canonical divisor and $χ(S, O)$ is the holomorphic Euler characteristic

Generalizations and Extensions

• The Riemann-Roch theorem can be generalized to higher-dimensional varieties using the Hirzebruch-Riemann-Roch theorem, which involves the Todd class and Chern character
• The Grothendieck-Riemann-Roch theorem extends the Riemann-Roch theorem to proper morphisms between smooth quasi-projective varieties, relating the pushforward of the Chern character of a coherent sheaf to the Chern character of its direct image sheaf
• The Atiyah-Singer index theorem is a far-reaching generalization of the Riemann-Roch theorem to the context of elliptic operators on compact manifolds, connecting analysis, geometry, and topology

Applying Riemann-Roch to Linear Systems

Computing Dimensions of Complete Linear Systems

• The Riemann-Roch theorem can be used to calculate the dimension of the complete linear system $|D|$ associated with a divisor $D$ on a curve or surface
• For a curve $C$ and a divisor $D$, the dimension of $|D|$ is given by $dim |D| = dim H⁰(C, O(D)) - 1$, which can be computed using the Riemann-Roch theorem for curves (e.g., for a divisor of degree $d$ on a curve of genus $g$, $dim |D| = d - g$ if $d \geq g$)
• For a surface $S$ and a divisor $D$, the dimension of $|D|$ is given by $dim |D| = h⁰(S, O(D)) - 1$, where $h⁰(S, O(D)) = χ(S, O(D)) + h¹(S, O(D)) - h²(S, O(D))$ can be computed using the Riemann-Roch theorem for surfaces and the Serre duality

Existence of Special Divisors

• The Riemann-Roch theorem can be used to determine the existence of special divisors, such as canonical divisors, anti-canonical divisors, or divisors with prescribed properties
• For a curve $C$ of genus $g$, the canonical divisor $K$ has degree $2g-2$, and the Riemann-Roch theorem implies that $dim |K| = g - 1$ (e.g., for a curve of genus 3, the canonical divisor defines an embedding into $\mathbb{P}^2$)
• For a surface $S$, the Riemann-Roch theorem can be used to study the existence of curves with prescribed properties, such as curves with a given genus or degree with respect to a fixed divisor (e.g., the existence of a rational curve on a K3 surface)

Geometric Consequences of Riemann-Roch

Canonical Embeddings and Kodaira Dimension

• For a curve $C$ of genus $g$, the canonical divisor $K$ leads to the existence of the canonical embedding of $C$ into projective space $\mathbb{P}^{g-1}$, which is a fundamental tool in the study of curve geometry
• The Kodaira dimension of a surface can be determined using the Riemann-Roch theorem, which provides information about the growth of the dimensions of the spaces of global sections of multiples of the canonical divisor
• Surfaces with Kodaira dimension $-\infty$ (ruled surfaces), $0$ (K3 surfaces, abelian surfaces), $1$ (elliptic surfaces), and $2$ (surfaces of general type) can be distinguished by the behavior of the pluricanonical series $|nK|$ for $n \geq 1$

Existence of Special Curves and Divisors

• The Riemann-Roch theorem can be used to prove the existence of special divisors, such as the anti-canonical divisor on a del Pezzo surface or the existence of a base-point-free pencil on a surface of Kodaira dimension 0
• For a surface $S$, the Riemann-Roch theorem can be used to study the existence of curves with prescribed properties, such as curves with a given genus or degree with respect to a fixed divisor
• The existence of special curves on surfaces, such as (-1)-curves on del Pezzo surfaces or elliptic curves on K3 surfaces, can be studied using the Riemann-Roch theorem and its consequences

Riemann-Roch vs Euler Characteristic

Definition and Properties of Euler Characteristic

• The Riemann-Roch theorem relates the Euler characteristic of a line bundle to its degree and the genus of the curve or the canonical divisor of the surface
• For a curve $C$ and a line bundle $L$, the Euler characteristic $χ(C, L)$ is defined as $χ(C, L) = dim H⁰(C, L) - dim H¹(C, L)$
• For a surface $S$ and a line bundle $L$, the Euler characteristic $χ(S, L)$ is defined as $χ(S, L) = h⁰(S, L) - h¹(S, L) + h²(S, L)$, where $h^i(S, L) = dim H^i(S, L)$
• The Euler characteristic is additive in short exact sequences of sheaves, which allows for the computation of the Euler characteristic of a line bundle in terms of the Euler characteristics of simpler line bundles

Riemann-Roch and Cohomology Dimensions

• The Riemann-Roch theorem states that for a curve $C$ and a line bundle $L$, $χ(C, L) = deg(L) - g + 1$, where $g$ is the genus of $C$
• For a surface $S$ and a line bundle $L$, the Riemann-Roch theorem states that $χ(S, L) = 1/2 c₁(L)·(c₁(L) - c₁(K)) + χ(S, O)$, where $c₁$ denotes the first Chern class and $K$ is the canonical divisor
• The Riemann-Roch theorem provides a powerful tool for computing the Euler characteristic of a line bundle and, consequently, the dimensions of the cohomology groups associated with the line bundle (e.g., for a curve of genus $g$ and a line bundle $L$ of degree $d$, $h⁰(C, L) = d + 1 - g$ if $d \geq 2g - 1$)
• Combined with vanishing theorems (such as the Kodaira vanishing theorem) and duality theorems (such as the Serre duality), the Riemann-Roch theorem allows for the computation of the dimensions of cohomology groups in various situations, providing valuable information about the geometry of curves and surfaces