4.3 Linear systems and complete linear systems

Linear systems are sets of effective divisors linearly equivalent to a given divisor on a curve. They're crucial for studying curve geometry and embeddings into projective spaces. Complete linear systems are projective spaces whose dimensions are given by the Riemann-Roch theorem.

The Riemann-Roch theorem relates a linear system's dimension to the divisor's degree and the curve's genus. This powerful tool helps compute dimensions and study curve geometry. Base-point free linear systems define morphisms to projective spaces, while base loci require blow-ups for analysis.

Linear systems and complete linear systems

Definition and properties

• A linear system associated to a divisor $D$ on a curve $C$ is the set $|D|$ of all effective divisors linearly equivalent to $D$
  • Two divisors $D$ and $D'$ are linearly equivalent if their difference $D - D'$ is the divisor of a rational function on $C$
  • Example: If $D = P + Q$ and $D' = R + S$ for points $P, Q, R, S$ on $C$, then $D$ and $D'$ are linearly equivalent if there exists a rational function $f$ on $C$ such that $div(f) = D - D' = P + Q - R - S$
• The complete linear system $|D|$ is a projective space whose dimension is given by the Riemann-Roch theorem
  • The elements of $|D|$ correspond to hyperplane sections of $C$ embedded into projective space via the map given by the divisor $D$
  • Example: If $D = P + Q + R$ is a divisor of degree 3 on a curve $C$, then $|D|$ is a projective space of dimension 2 (a projective plane), and its elements correspond to lines in the projective plane that intersect $C$ at the points $P, Q, R$

Complete linear systems as projective spaces

• The complete linear system $|D|$ is a projective space of dimension $r = dim |D|$
  • If $D$ is a very ample divisor, then $|D|$ gives an embedding of the curve $C$ into the projective space of dimension $r$
  • The degree of the embedding is equal to the degree of the divisor $D$
    • Example: If $D$ is a divisor of degree 3 on a curve $C$ of genus 1, then $|D|$ is a projective plane, and the embedding of $C$ into this plane is a cubic curve
• If $D$ is a canonical divisor $K$, then $|K|$ is called the canonical system and its dimension is the genus $g$ of the curve $C$
  • The canonical system plays a crucial role in the study of curves and their geometry
  • Example: For a curve $C$ of genus 2, the canonical divisor $K$ has degree 2, and the canonical system $|K|$ is a projective line (a pencil of lines) that maps $C$ to a double cover of $\mathbb{P}^1$

Properties of complete linear systems

Riemann-Roch theorem and dimension computation

• The Riemann-Roch theorem states that for a divisor $D$ on a curve $C$ of genus $g$, the dimension of the complete linear system $|D|$ is given by $dim |D| = deg(D) - g + 1 + dim |K - D|$, where $K$ is a canonical divisor
  • If $deg(D) \geq 2g - 1$, then $dim |K - D| = 0$, and the dimension formula simplifies to $dim |D| = deg(D) - g + 1$
  • Example: For a curve $C$ of genus 2 and a divisor $D$ of degree 4, the Riemann-Roch theorem gives $dim |D| = 4 - 2 + 1 = 3$, so $|D|$ is a projective space of dimension 3
• The theorem relates the dimension of a linear system to the degree of the divisor and the genus of the curve
  • It provides a powerful tool for computing the dimension of linear systems and studying the geometry of curves
  • Example: If $C$ is a curve of genus 3 and $D$ is a divisor of degree 5, then $dim |D| = 5 - 3 + 1 = 3$, while if $D$ has degree 2, then $dim |D| = 2 - 3 + 1 + dim |K - D| = dim |K - D|$, which depends on the specific divisor $D$

Base loci and base-point free linear systems

• The base locus of a linear system $|D|$ is the set of points $P$ on the curve $C$ such that $P$ is contained in every divisor in $|D|$
  • Geometrically, the base locus corresponds to the intersection of all hyperplanes in the projective space $|D|$ that contain the image of $C$ under the embedding given by $D$
  • Example: If $D = P + Q + R$ and all divisors in $|D|$ contain the point $P$, then $P$ is a base point of the linear system $|D|$
• If the base locus is empty, then $|D|$ is called base-point free, and it defines a morphism from $C$ to a projective space of dimension $dim |D|$
  • The degree of this morphism is equal to the degree of the divisor $D$
  • Example: If $D$ is a very ample divisor on $C$, then $|D|$ is base-point free and defines an embedding of $C$ into a projective space
• Base points can be studied by considering the blow-up of the curve at those points, which modifies the linear system and its dimension
  • Blowing up a base point increases the dimension of the linear system by 1 and decreases the degree of the divisor by 1
  • Example: If $D = 2P + Q$ has a base point at $P$, then blowing up $C$ at $P$ gives a new curve $C'$ and a new linear system $|D' = Q|$ on $C'$, where $D'$ is the strict transform of $D$ under the blow-up

Dimension of complete linear systems

Computation using the Riemann-Roch theorem

• The Riemann-Roch theorem provides a formula for computing the dimension of a complete linear system $|D|$ on a curve $C$ of genus $g$
  • $dim |D| = deg(D) - g + 1 + dim |K - D|$, where $K$ is a canonical divisor
  • If $deg(D) \geq 2g - 1$, then $dim |K - D| = 0$, and the formula simplifies to $dim |D| = deg(D) - g + 1$
  • Example: For a curve $C$ of genus 4 and a divisor $D$ of degree 6, the Riemann-Roch theorem gives $dim |D| = 6 - 4 + 1 = 3$
• The dimension of a linear system can be used to study the geometry of the curve and its embeddings into projective spaces
  • If $dim |D| = r$, then $|D|$ gives a map from $C$ to a projective space of dimension $r$
  • The degree of this map is equal to the degree of the divisor $D$
  • Example: If $C$ is a curve of genus 2 and $D$ is a divisor of degree 5, then $dim |D| = 5 - 2 + 1 = 4$, so $|D|$ gives a map from $C$ to a projective space of dimension 4 (a 4-dimensional projective space)

Special linear systems

• Certain linear systems have special properties and play important roles in the study of curves
  • The canonical system $|K|$, where $K$ is a canonical divisor, has dimension equal to the genus $g$ of the curve
    • Example: For a curve $C$ of genus 3, the canonical system $|K|$ is a projective plane that maps $C$ to a plane quartic curve
  • Multiples of the canonical system, such as $|2K|$ or $|3K|$, give higher-dimensional embeddings of the curve and are related to the geometry of the moduli space of curves
    • Example: For a curve $C$ of genus 2, the linear system $|2K|$ is a projective space of dimension 3 that maps $C$ to a surface of degree 4 in $\mathbb{P}^3$ (a quartic surface)
• The study of special linear systems and their dimensions is a central topic in the theory of algebraic curves and their moduli spaces
  • The dimensions of these linear systems are related to important invariants of the curve, such as its gonality and Clifford index
  • Example: A curve $C$ is hyperelliptic if and only if it has a linear system of degree 2 and dimension 1 (a $g^1_2$), which gives a double cover of $\mathbb{P}^1$

Geometric interpretation of linear systems

Embeddings and maps to projective spaces

• A linear system $|D|$ on a curve $C$ defines a map from $C$ to a projective space of dimension $dim |D|$
  • If $|D|$ is base-point free, then this map is a morphism (a well-defined map between algebraic varieties)
  • The degree of the map is equal to the degree of the divisor $D$
    • Example: If $D$ is a divisor of degree 4 on a curve $C$ of genus 2, then $|D|$ gives a map from $C$ to a projective space of dimension 3, and the image of $C$ under this map is a curve of degree 4 in $\mathbb{P}^3$
• If $D$ is a very ample divisor, then $|D|$ gives an embedding of $C$ into a projective space
  • An embedding is a morphism that is injective (one-to-one) and has injective differential at every point
  • Example: If $C$ is a curve of genus 3 and $D$ is a divisor of degree 6, then $|D|$ is a very ample linear system that embeds $C$ into a projective space of dimension 5 as a curve of degree 6

Base loci and blow-ups

• The base locus of a linear system $|D|$ consists of the points on the curve $C$ that are contained in every divisor in $|D|$
  • Geometrically, the base locus corresponds to the intersection of all hyperplanes in the projective space $|D|$ that contain the image of $C$
  • Example: If $D = 2P + Q + R$ for points $P, Q, R$ on $C$, then $P$ is a base point of $|D|$ because every divisor in $|D|$ contains $P$ with multiplicity at least 2
• To remove base points and obtain a base-point free linear system, one can perform a blow-up of the curve at the base points
  • Blowing up a point $P$ on $C$ replaces $P$ with a projective line (the exceptional divisor) and modifies the linear system accordingly
  • The blow-up increases the dimension of the linear system by 1 and decreases the degree of the divisor by the multiplicity of the base point
    • Example: If $D = 3P + Q$ has a base point at $P$ with multiplicity 3, then blowing up $C$ at $P$ gives a new curve $C'$ and a new linear system $|D' = Q|$ on $C'$, where $D'$ is the strict transform of $D$ under the blow-up
• The study of base loci and blow-ups is important for understanding the geometry of linear systems and their associated maps and embeddings
  • Blowing up base points can simplify the linear system and make it base-point free, which allows for a more straightforward analysis of the associated map or embedding
  • Example: If $D = 2P + 2Q$ has base points at $P$ and $Q$, then blowing up $C$ at both points gives a new curve $C'$ and a base-point free linear system $|D'|$ on $C'$, where $D'$ is the strict transform of $D$ minus the exceptional divisors over $P$ and $Q$