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The dimension of a complete linear system

Statement

Assume the Axiom of Choice as inherited from proper-cohomology, projective-space and Riemann-Roch suppliers (The Axiom of Choice). Let $k$ be a field, let $C$ be a smooth proper geometrically integral curve over $k$ (Curves over a field) of genus $g = g (C)$ (Genus via the Euler characteristic), let $D$ be a divisor on $C$ (Divisors on a smooth proper curve) and let $∣ D ∣$ be its complete linear system (Complete linear system). For a finite-dimensional $k$ -vector space $W$ , write $P_{k} (W) : = Proj ({Sym}_{k} (W^{\lor}))$ for the projective scheme parameterizing one-dimensional subspaces of $W$ . When $∣ D ∣$ is nonempty, put $P_{D} : = P_{k} (L (D)) .$ The section-divisor correspondence identifies the set $∣ D ∣$ with the set of $k$ -rational points $P_{D} (k)$ .

Then:

$∣ D ∣$ is nonempty if and only if $l (D) \geq 1$ (The Riemann-Roch dimension l(D)); equivalently $∣ D ∣$ is empty exactly when $L (D) = 0$ ;
if $∣ D ∣$ is nonempty and $l (D) = r \geq 1$ , a choice of basis of $L (D)$ identifies $P_{D}$ with the projective scheme $P_{k}^{r - 1}$ , and we define $\dim_{k} ∣ D ∣ : = \dim P_{D}$ . Its dimension is $\dim_{k} ∣ D ∣ = l (D) - 1 = \deg_{k} (D) - g + i (D),$ with $i (D)$ the index of speciality (The index of speciality i(D));
if $D$ is nonspecial (Special and nonspecial divisors) then $∣ D ∣$ is nonempty if and only if $\deg_{k} (D) \geq g$ , and in that case $\dim_{k} ∣ D ∣ = \deg_{k} (D) - g$ ;
for the zero divisor, $∣ 0 ∣ = {0}$ is a single $k$ -rational point, $P_{0} ≅ P_{k}^{0}$ , and $\dim_{k} ∣ 0 ∣ = 0 = l (0) - 1$ . The nonspecial value $\deg_{k} (0) - g = - g$ is attained as $\dim_{k} ∣ 0 ∣$ exactly when $g = 0$ .

Thus $\dim_{k} ∣ D ∣$ means the Krull dimension of the projectivization scheme, not the dimension of its set of $k$ -rational points. No algebraic-closure hypothesis on $k$ is used.

Facts & Assumptions

Given: the Axiom of Choice; a field $k$ ; a smooth proper geometrically integral curve $C$ over $k$ of genus $g = g (C)$ ; a divisor $D$ on $C$ ; the complete linear system $∣ D ∣$ ; and the index of speciality $i (D) = h^{1} (C, O_{C} (D))$ .

[F1]

The assignment $f \mapsto div (f) + D$ induces a bijection $(L (D) ∖ {0}) / k^{\times} \to ∣ D ∣$ . Hence $∣ D ∣$ is identified with the set of $k$ -lines in $L (D)$ and is empty exactly when $L (D) = 0$ (Effective divisors linearly equivalent to D are sections modulo scalars, Complete linear system).

[F2]

Under the stated Axiom of Choice, the current Finite-dimensionality of the Riemann-Roch space proves that $L (D)$ and all $H^{q} (C, O_{C} (D))$ are finite-dimensional. Thus $l (D)$ and $i (D)$ are nonnegative integers (The Riemann-Roch dimension l(D), The index of speciality i(D), Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis). For $D = 0$ , $H^{0} (C, O_{C}) = k$ , so $l (0) = 1$ (Functions on a proper curve).

[F3]

The cohomological Riemann-Roch identity is $l (D) - i (D) = \deg_{k} (D) + 1 - g$ , and a divisor is nonspecial exactly when $i (D) = 0$ (Riemann-Roch as l minus i, Special and nonspecial divisors).

[F4]

For a nonzero finite-dimensional $k$ -vector space $W$ of dimension $r$ , $P_{k} (W) : = Proj ({Sym}_{k} (W^{\lor}))$ is the projective scheme parameterizing lines in $W$ . A basis of $W$ identifies it with $P_{k}^{r - 1}$ ; its $k$ -rational points are exactly the one-dimensional $k$ -subspaces of $W$ : if $e_{0}, \dots, e_{r - 1}$ is a basis, the points $[a_{0} : \dots : a_{r - 1}]$ with $a_{i} \in k$ and not all $a_{i} = 0$ , modulo common nonzero scalar, correspond to the line spanned by $\sum_{i} a_{i} e_{i}$ . The case $r = 1$ is $P_{k}^{0}$ (Symmetric algebra of a vector space, Relative projective space from standard charts, Projective space is Proj of a polynomial ring).

[F5]

The projective scheme $P_{k}^{r - 1}$ has $r$ standard open charts, each isomorphic to $Spec k [y_{1}, \dots, y_{r - 1}]$ (Relative projective space from standard charts, Projective space is Proj of a polynomial ring). The coordinate ring has Krull dimension $r - 1$ (A polynomial ring in n variables over a field has dimension n). Since a field is Noetherian, these polynomial rings are Noetherian (A field has only the zero ideal and itself, hence is Noetherian, Every algebra of finite type over a Noetherian ring is a Noetherian ring), and their spectra are Noetherian spaces (The spectrum of a Noetherian ring is a Noetherian topological space). Any descending chain of closed subsets of $P_{k}^{r - 1}$ stabilizes after restriction to each standard chart. Since there are finitely many charts, the maximum of their stabilization indices works on the whole projective space, so it is Noetherian. Its dimension is the supremum of the chart dimensions by Dimension can be computed on an open cover and Chain dimension and the empty-space convention. Consequently $\dim P_{k}^{r - 1} = r - 1$ for every field $k$ .

[F6]

The Axiom of Choice is inherited from the finiteness, projective-space and Riemann-Roch suppliers. The only choice made below is a basis of the finite-dimensional space $L (D)$ (The Axiom of Choice).

[F7]

The $k$ -degree is $\deg_{k} D = \sum_{x} n_{x} [κ (x) : k]$ , a sum over the finite support of $D = \sum_{x} n_{x} [x]$ ; the zero divisor has empty support and $\deg_{k} (0) = 0$ (Degree divisor proper curve, Divisors on a smooth proper curve).

Proof

technique · read the complete linear system off the bijection with the $k$-lines in $L(D)$, use its projectivization scheme for dimension, and substitute the Riemann-Roch identity

1.1F1F2

The empty case. By [F1], $∣ D ∣$ is in bijection with the $k$ -lines in $L (D)$ . It is empty exactly when $L (D) = 0$ , which by [F2] is equivalent to $l (D) = 0$ . Therefore $∣ D ∣$ is nonempty exactly when $l (D) \geq 1$ .

1.2F1F2F4F5

The projective parameter scheme. Suppose $∣ D ∣$ is nonempty, so $L (D) \neq 0$ by [F1]. By [F2], $L (D)$ is finite-dimensional; put $r = l (D) = \dim_{k} L (D) \geq 1$ . Choose a basis. By [F4] it identifies $P_{D} = Proj ({Sym}_{k} (L (D)^{\lor}))$ with $P_{k}^{r - 1}$ , whose $k$ -rational points are the $k$ -lines in $L (D)$ . By [F1], these points are exactly $∣ D ∣$ . The standard-chart calculation [F5] then gives $\dim_{k} ∣ D ∣ : = \dim P_{D} = r - 1 = l (D) - 1.$

1.3F1F2F3F4F5F7

The zero divisor. By [F2], $L (0) = H^{0} (C, O_{C}) = k$ and $l (0) = 1$ . Its unique $k$ -line maps under [F1] to $div (1) + 0 = 0$ , so $∣ 0 ∣ = {0}$ and $P_{0} ≅ P_{k}^{0}$ by [F4]. Therefore $\dim_{k} ∣ 0 ∣ = 0 = l (0) - 1$ . By [F7], $\deg_{k} (0) = 0$ , and by [F3], $i (0) = g$ . The nonspecial value $\deg_{k} (0) - g = - g$ equals the actual dimension zero exactly when $g = 0$ .

2.1F3step 1.2

Riemann-Roch substitution. By [F3], $l (D) = \deg_{k} (D) + 1 - g + i (D)$ . Combining this with step 1.2 gives $\dim_{k} ∣ D ∣ = l (D) - 1 = \deg_{k} (D) - g + i (D) .$

3.1F3step 1.1step 2.1

The nonspecial case. If $D$ is nonspecial then $i (D) = 0$ by [F3], so $l (D) = \deg_{k} (D) + 1 - g$ . By step 1.1, $∣ D ∣$ is nonempty exactly when this integer is at least one, equivalently when $\deg_{k} (D) \geq g$ . When this holds, step 2.1 gives $\dim_{k} ∣ D ∣ = \deg_{k} (D) - g$ .

4.1F1F2F3F4F5F6F7step 1.1step 1.2step 2.1step 3.1step 1.3

Conclusion and choice accounting. Steps 1.1 and 1.2 give $∣ D ∣ \neq \emptyset ⟺ l (D) \geq 1$ and define its dimension as the Krull dimension of the projectivization scheme. Steps 2.1 and 3.1 give the general and nonspecial dimension formulas; step 1.3 verifies all four zero-divisor claims. The projective-space dimension computation uses standard relative Proj charts and holds over every field, including fields that are not algebraically closed. Choice is used only for the cohomology and Riemann-Roch suppliers and the basis of $L (D)$ in step 1.2.

5.1F1F2F3F4F5F6F7∎

Current supplier boundary. The section-to-divisor set bijection is given by the current Effective divisors linearly equivalent to D are sections modulo scalars and Complete linear system. The Cartier, divisor-space, finite-dimension and Riemann-Roch sources cited above are present in the working tree; their draft or published status remains as recorded in their frontmatter. The projective parameter scheme is defined in this item and its dimension is proved from the published relative-Proj charts and affine polynomial-ring dimension theorem. The construction and dimension calculation apply over arbitrary fields $k$ .

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Sources

William Fulton, Algebraic Curves (Internet Archive copy), Chs. 8 and 6 (standard reference, not scraped)
Michael Artin, MIT 18.721 Introduction to Algebraic Geometry (July 20, 2020 notes), Ch. 8 (standard reference, not scraped)
The Stacks Project, Algebraic Curves (tag 0BRV) (standard reference, not scraped)