A linear system of divisors is a fundamental construct in algebraic geometry that parametrizes a family of effective divisors on an algebraic variety (or scheme) which are linearly equivalent to a given divisor. Formally, let $X$ be a projective variety over an algebraically closed field and let $D$ be a divisor on $X$. The complete linear system associated with $D$ is the projective space
$$ |D| = \mathbb{P}\bigl(H^{0}(X, \mathcal{O}_{X}(D))\bigr), $$
where $H^{0}(X, \mathcal{O}{X}(D))$ denotes the vector space of global sections of the line bundle $\mathcal{O}{X}(D)$ (the sheaf associated to $D$). Each point of $|D|$ corresponds to an effective divisor linearly equivalent to $D$; two sections that differ by a non‑zero scalar define the same divisor.
Key aspects
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Linear equivalence – Two divisors $D_1$ and $D_2$ are linearly equivalent, written $D_1 \sim D_2$, if there exists a rational function $f$ on $X$ such that $D_1 - D_2 = \operatorname{div}(f)$. A linear system collects all effective representatives of a given linear equivalence class.
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Complete vs. incomplete systems – The complete linear system $|D|$ contains all effective divisors linearly equivalent to $D$. An incomplete (or linear) subsystem is a projective subspace $$ \mathbb{P}(V) \subseteq |D| $$ where $V \subseteq H^{0}(X, \mathcal{O}_{X}(D))$ is a linear subspace of sections. Such subsystems are often denoted by a lowercase $g$-notation, e.g., $g^r_d$ on a smooth curve, indicating a linear system of dimension $r$ and degree $d$.
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Base locus – The set of points of $X$ that belong to every divisor in the linear system. For a subsystem corresponding to $V$, the base locus equals the common zero set of all sections in $V$.
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Morphisms associated to linear systems – A base‑point‑free linear system defines a morphism $$ \phi_{|D|} : X \longrightarrow \mathbb{P}^{r}, $$ where $r = \dim |D|$. The morphism maps a point $x \in X$ to the evaluation of the sections at $x$, interpreted in projective coordinates. If the system has base points, one obtains a rational map instead.
Classical examples
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Projective space $\mathbb{P}^n$ – The hyperplane linear system $|\mathcal{O}_{\mathbb{P}^n}(1)|$ consists of all hyperplanes; it is identified with the dual projective space $(\mathbb{P}^n)^{\vee}$.
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Curves – On a smooth projective curve $C$ of genus $g$, a divisor $D$ of degree $d$ yields a linear system $g^r_d$ with $r = \ell(D)-1$, where $\ell(D) = \dim H^{0}(C, \mathcal{O}_C(D))$. The Riemann–Roch theorem relates $r$, $d$, and $g$.
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Canonical linear system – The divisor associated to the canonical line bundle $K_X$ gives the canonical linear system $|K_X|$, which plays a central role in classification theory, e.g., the canonical map of a surface of general type.
Applications
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Embedding criteria – Very ample divisors produce complete linear systems that embed $X$ into projective space. The Nakai–Moishezon and Kleiman criteria are expressed in terms of positivity of divisor classes, often verified using linear systems.
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Construction of moduli – Families of curves or higher‑dimensional varieties with prescribed linear systems are used to define and study moduli spaces (e.g., the moduli of $g^r_d$ on curves).
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Resolution of singularities – Blow‑ups are often described via linear systems: the exceptional divisor corresponds to the base locus of a linear system that has been resolved.
Related concepts
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Picard group $\operatorname{Pic}(X)$ – The group of line bundles modulo linear equivalence; linear systems correspond to projective subspaces of $\operatorname{Pic}(X)$.
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Linear series – Synonymous with linear system, especially in the context of curves.
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Movable linear system – A linear system whose base locus has codimension at least two; such systems are important in the Minimal Model Program.
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Linear equivalence class – The set of all divisors linearly equivalent to a given divisor; the complete linear system is the subset consisting of effective members.
References
- Hartshorne, R. Algebraic Geometry, Graduate Texts in Mathematics 52, Springer, 1977. Chapters II–IV.
- Griffiths, P., Harris, J. Principles of Algebraic Geometry, Wiley, 1978.
- Eisenbud, D., Harris, J. The Geometry of Schemes, Springer, 2000.
These sources provide detailed treatments of linear systems of divisors, including their cohomological interpretation, base loci, and applications to projective embeddings.
